EDIT: Wow, thank you for all the amazing answers, discussion and for the links! I will definitely go through them all during the weekend :)
To add onto the modern military aspect that another has touched upon...
In modern war between major world powers or even just major regional ones, castles would be useless for anything more than a command post and possibly a distribution point. Quit a few castles overlooked routes and positions that are still valued and used to this day.
Their defensive capabilities would be all but useless against more than a token amount of modern air support or artillery bombardment. The Germans effectively proved the futility of trying to plan national defenses around heavily ingrained, static fortifications in the first World War. Belgium had some of the strongest and most advanced fortresses in the world when Germany invaded. Most of them had been constructed in the second half of the 19th century, some only 2-3 decades prior, and they had been specifically engineered to take a beating from the most powerful artillery of their day. They were magnificent bulwarks of concrete and pill boxes, defended by thousands of men, and constructed in such a way that they could provide support for one another.
Do you know what the Germans did? They brought bigass guns. The biggest in history up to that point. Guns that dwarfed the ones that these forts had been designed to handle.
Within hours of these monster mortars opening up on them most of these forts surrendered. They had held out amazingly well against the initial waves of infantry the Germans had thrown at them (as they should have). However modern technology rendered them useless almost immediately.
A castle, even built with modern materials and design wouldn't last more than a few minutes of heavy bombardments from JDAM's: https://www.youtube.com/watch?v=7fc8bgf1pGk&ab_channel=EpigeneticAlteration
Yet the Belgians didn't fully get the message from this war. In the interwar period from 1932-1935 they built a monstrosity on the border with Germany called Fort Eben-Emael. This thing was almost certainly the strongest static defensive position on the planet when it was built. To ensure Belgium was neutralized, Germany had to plan its invasion around dealing with this place at the very start. Which they did.
They pulled off what I can honestly say is easily one of the ballsiest and daring special operations of modern times. A 600 An 82 man force of SS flew into the fort via glider (so they were silent) then cut, shot, and exploded their way to several critical points in the fort that would cripple its capabilities and allow the German army to pass into Belgium unmolested. At the time there were 1,200 defenders in the complex. There should have been at least 3 times that.
3 years of work and the equivent of billions spent on this thing; the cornerstone of Belgium's national defense, and it was neutralized in a night at the cost of just 6 enemy lives. So not only are fortresses obsolete in modern combat due to ludicrous advances in military firepower, but also because with modern tech and tactics they can be infiltrated far to easily to reliably plan a national defense around.
The closest things to castles/fortresses that modern nation states still build are things like the Pentagon or NORAD. Places not really designed to weather a siege, but possibly one to several bigass explosions (a short ferocious missile bombardment or a nuclear strike).
So that's how castles stand with first rate powers.
Developing/third world regions are a somewhat different story though. In places with limited to no air to ground factors and an inability to supply and transport large amounts of standardized artillery, let alone heavy armor, castles and the positions they occupy can still be somewhat valuable in a variety of situations (at least for a limited time).
A day in a castle largely depended on who you were, and what time period you lived in. For this I will be keeping my answers to the 13th century onward. At the very top you had the lord and lady of the castle who would range in titles from baron upwards. (Knights generally held manors rather than castles.) The lord would be running the estate, hunting, feasting or any number of tasks his status would require him to preform. The next important people were the lord's stewards and his marshall. There was an estate steward, helped by bailiffs, whose task was to make sure all taxes, fines and rents were paid on time. Due to most taxes from peasants being paid in food, this was especially important to feed the castle's staff. Then there was the house steward, whose job was to feed the castle, from servant to lord. There were clerks who assisted the stewards. The chamberlain was in charge of a lord's personal wealth and his chambers, hence the name. These people were the bureaucracy of a castle, and would run the day to day functions.
Then there were indoor staff attached to any number of rooms. The kitchen had four senior roles: the Pantler and Butler in charge of food and drink, the chief cook in charge of the cooks and kitchen assistants, and the Ewerer in charge of the table linen. There were waiters and cleaners, tailors, seamstresses, nursemaids and the lady's maids. These people had specific tasks related to the running of the castle and would eat and live separately from the important people.
There were handymen, whose job it was to fix the castle and keep up on repairs. These men were separate from the skilled craftsmen, like the blacksmith, and were mostly manual labor.
The outdoor servants were overseen by the marshall. The Marshall oversaw hunting and the animals as well as all the staff assigned to do tasks outdoors. The variety on the outdoor staff depended on how large the castle was and what other outbuildings it had.
The warriors of the castle were the men-at-arms, under the command of the constable. They would see to the security of the castle and lord.
Dependent on how important the castle was, their was usually a chaplain, but could house someone more important to the church hierarchy.
These people would go about their lives and jobs depending on their station, so a day in the castle would vary much on your station and what the lord decided to do that day.
As to whether a castle would be useful in a 21st century war; as a fortification it would not. With the rise of cannons and other forms of gunpowder artillery, the high stone walls couldn't stand and there was a shift to the iconic star fortifications. With modern artillery, the castle would simply be blasted apart. A castle also is a prime target for aircraft, since it's an above ground target. The defining feature of modern fortifications is lots of materials that can absorb impacts, like sand and earth. Concrete is used to brace these materials to stop the slowed down projectiles.
A good source for the castle material, though geared towards children (also the only source I have handy), is the "The Usborne Book of Castles" by Lesley Sims.
I found that there are hundreds of castles from 12th-17th centuries in England alone.
Wikipedia has a useful list at https://en.wikipedia.org/wiki/List_of_castles_in_England, along with some interesting discussion of exactly what constitutes a "castle".
What does a "day in the life of such castle" look like?
Do you mean now, or at the time they were built? There are certainly lots of sources on the latter, but of course you're talking over five centuries so things will change. If you want the ELI5 version, there's a nice article here.
If the former, it will really depend - some castles are privately owned (Sheriff Hutton, near me, is nothing more than a few picturesque lumps of masonry in a field) and nothing much happens there beyond basic maintenance. At the other end of the scale, others such as Harlech or Conwy in Wales are managed by English Heritage and are tourist attractions with visitor interpretation so will have different things going on. Others such as Norwich have been converted into museums and the like. Durham castle houses student accommodation for the University of Durham ... the list goes on.
Are people living in them?
The only inhabited castle of that era I can think of off the top of my head is Windsor, which has of course been heavily remodelled several times over the centuries since. There may be others, but it's a vanishingly small number.
Would they still be useful in a war of 21st century?
No.
These castles have been obsolete as military fortifications since cannon came into widespread use during the 15th century - the French army with a big artillery train went through the Italian peninsula like a hot knife through butter in the 1490s and really proved it.
England was a bit unusual in that it was relatively stable and centralised after the Wars of the Roses (and unlikely to be invaded from abroad), and so there wasn't really much need for fortifications after that. As soon as nobles didn't need to live in cold, draughty castles for defence, they either knocked them down and rebuilt them, or heavily converted them into places that may have been described as "Castles" but were basically stately homes.
About the only military use I can think of for a medieval style castle in the modern day is in some sort of guerilla-style war, where the only opponents are human beings with small arms. Even then, though, something as simple as a mortar would render the position indefensible quite quickly.
I realise that the above is not a detailed, fully sourced answer in the highest traditions of /r/AskHistorians, but hopefully it will act as a way to tease out exactly what OP is asking and allow more detailed, specific answers.
Although I'm in Scotland I assume this still applies to me unless you really just meant England and not the UK?
There is a 14th century tower in my next door neighbours garden. It isn't very impressive, it looks like this now:
But it has a lot of history, hundreds of years of history and it is a beautiful place to just sit and think.
It was used as a hunting lodge for Royalty at one time and in the fields opposite I find lots of musket balls with my metal detector.
The doorway is still intact and gives you a real indication of how big the tower used to be. The walls are incredibly thick,many feet thick and you can still see very large square holes in the walls either side of the door archway where they put the wooden beam (or whatever they used) to close it.
The surrounding topography tells a story - it is flanked on one side by a river and the other sides by a moat which has long since dried up.
It was an important location back when the English were fighting the Scottish and the Scottish were fighting the Scottish.
Would they still be useful in a war of 21st century?
This question has drawn a huge focus on the technologies of war, particularly heavy artillery and (surprisingly!) gliders. Certainly there are technological means for overcoming castles now, much as there were in the 12th-17th centuries. And yet castles were built in the middle ages and early modern period, and some are still occupied now. Why?
Let's take the Danish Kastellet in Copenhagen as an example. It's construction began in 1626, it was twice defended (1801 and 1807, although in both cases the battle was decided at sea without an assault on the castle itself), it has been mostly converted to park space (complete with windmill and church), and yet it retains critical military functions.
Does this make sense? I'd argue that it absolutely makes sense from at least three perspectives.
First, although there is public access to the park, the access remains restricted to two narrow roadways. (Yes, you can tour the castle using Google Street View.) This makes surveillance easy, and in the case of a perceived threat, it would require little effort to stop all access to the castle. In this way, the Kastellet still controls movement much as originally designed.
Second, the Kastellet is a famous cultural landmark, especially since its conversion to a park space with additional historic structures as well as modern sculpture. This will deter some potential attacks. Any assailant who wants to maintain international legitimacy would avoid attacking a cultural site, as evidenced most recently in the uproar over the IS occupation of Palmyra.
This limits potential threats to parties who are unconcerned with international legitimacy (read: terrorists), who would be just as likely to attack a cultural site as a military target. By collocating some of their high military functions with a significant cultural site, the Danes have reduced the number of sites they need to defend against these kinds of threats, which shows a wise economy of force.
Third, the Kastellet has enduring value as a historical seat of the Danish military. By maintaining military functions at the site, the Danish government inspires a sense of continuity of the past, which can help enrich a sense of shared history, critical to nationalism. This might seem minor, but it can be significant.
Imagine if, at the height of the Iraq war, Bush had transitioned military planning from the Pentagon to Austin, TX. In some ways this makes sense: Texas tends to be pro-military, it is the home of a tremendous concentration of military bases, it was a center of Bush support, and it was the headquarters of Halliburton. But imagine the kickback, if the traditional seat of American military power had been transferred to follow the needs of an unpopular conflict and expressed a bald reliance on non-governmental contractors. There are good reasons for states to maintain a presence in traditional places of power, and these can become critical during times of war.
In short, I'd say castles do still have the potential to be useful in 21st century wars, and sometimes for the very same reasons why they were first built: they restrict access, they carry cultural value, and they assert an entrenched presence in the national landscape. No, they won't hold up to big guns or paratroopers with shaped charges, but wars are never decided by weapons alone.
Yes, many of them are still occupied. Windsor has already been mentioned. The Tower of London and Edinburgh Castle are also well known. The latter is a military base: there are others such as Fort George in the Highlands, if you are interested in later ones.
Most inhabited castles, however, are private homes such as Raby Castle, which have normally been updated to make them more comfortable to live in. The smallest inhabited ones are pele towers in the formerly debatable lands between Scotland and England. These are fortified towers often built as adjuncts to farms. While they are generally too small to make a complete home, they are usually now part of the farmhouse building, providing part of the habitable space. They are not terribly expensive : if you can find one, it will probably not cost more than a normal farm-house in that region.
Would they be useful in war? Not the ones which are large enough to inhabit, but there are several thousand [pillbox fortlets](https://en.wikipedia.org/wiki/Pillbox_(military)) in the south of England, left over from WW II. These were expected to be useful in resisting a German invasion.