The history of gunpowder weapons is anything but a straightforward case of guns defeating knights and replacing other weapons. (All those years playing Civilisation, it was misleading me!)

I'm going to talk about handguns here, as they're most relevant. Most of this information comes from Weapons and Warfare in Renaissance Europe by Bert S Hall, which is a book I especially sought out because this subject puzzled me.

The first thing to remember is that up until the 19th century guns were mostly, slow firing, short-ranged and inaccurate. We're talking an effective range of not more than 100m (especially against armour), and a firing rate of at most 3-4 shots a minute (for experienced troops). Against charging cavalry you'll probably only get two shots at best, which is not likely to be enough to stop them. Even infantry could advance quickly enough to engage in melée combat (in fact, melée combat was regularly used by the British army well into the 19th century). According to Hall, musket technology did not advance very much between the 16th and 18th centuries.

So in many ways they are comparable to crossbows. Both weapons are pretty good at penetrating armour, but are slow to reload. So you really need a defensive position or supporting melée troops (pikemen) to use them effectively. Guns also have disadvantages, like being more expensive to build and operate, needing a lit taper, and being more sensitive to damp.

It's also wrong to think of guns as brilliant armour-piercing weapons. They were better than bows, but newer 'bullet proof' armour was developed as a counter. The term "musket" seems to have originally meant an especially large handgun intended to punch through heavy armour.

So in fact, guns and crossbows were fielded alongside one another for a long time, even being used in mixed formations. In England especially, the longbow remained important until at least the early 16th century. It took several centuries for guns to thoroughly replace bows. I believe this was due to marginal improvements in range and armour penetration but I haven't seen a definitive conclusion.

So for centuries we have the following situation:

Guns (and crossbows) are powerful weapons, but unless they're in a defensive position are vulnerable to cavalry.
So missile units need the support of melée units as protection. In most cases this meant pikemen, who were particularly strong against cavalry. The introduction of the bayonet in the 17th century turned musketeers into spearmen and after some time led to the demise of dedicated pike units. (I don't know exactly how long this took.)
Light cavalry were cheaper and more mobile, but could be defeated by more heavily armoured cavalry (ie. "knights"). Heavy cavalry was also more effective against infantry, and somewhat less vulnerable to missile fire.

This still leaves a question: why did full plate armour fall out of use? I don't know what the consensus on this is, but Hall argues it was down to the Reiter, already described by Hazzardevil. Using a multiple wheel-lock pistols at very short range (and ideally shooting the horses), these troops could defeat armoured cavalry. They also had the advantage of being a socially acceptable path for young nobles, as wheel-lock pistols were expensive and prestigious, while being considerably less expensive than maintaining a full suit of armour and a heavy warhorse.

Dedicated melee combat units in Western Europe until at least the end of the 30 Years war. "Cold Steel" regiments, units that fought on foot and primarily with a melee weapon, like swords and bucklers or Zweihanders were used to break open pike and shot formations.

Piles were a long spear that varied hugely in length at the time. The "half-pike" was favoured by the Parliamentarians and the New Model Army and was around 6 ft in length. On the continent likes varied and could end up over 5 metres long. These were more effective against cavalry and infantry with shorter pikes, but suffered against the aforementioned cold steel regiments, if they could reach hand to hand combat.

Pikes were mixed with muskets in various formations and over time phased out until most armies relied solely on the musket as the weapon of choice for the rank and file.

Cavalry last right up to WW1, with their roles changing. At the beginning of the Renaissance many armies, particularly France relied on Cavalry to win the battle, with everything else as a supporting element. These would have been the heavily armed and armoured "knights" you think of, but by this time the process of becoming a titled knight was so expensive few could afford it.

They typically used lances and a sword, with some areas favouring a different sidearm, such as the Germans who were using maces and the Polish Hussars occasionally carried axes. Melee weapons with guns built in were also popular, as were guns with blades or maceheads attached.

Other forms of cavalry arose during the Renaissance, "Reiters," a German word for Rider, referred to an armoured cavalryman with a pistol instead of a lance and performed a maneuver called the Carrocole. A block of cavalry woe ride forwards towards the enemy and the first rank would shoot and run around to the back to reload.

Opinion was divided at the time as to if this effective. Gustavus Adolphus Magnus (Magnus meaning "The Great") was opposed to this and notably among his light cavalry, the Finnish Hakkapells, had two pistols, an axe for out of combat use and a sword. They were made to charge rather than perform a carracole.

Other cavalry included dragoons, who fought on foot and rode to the battle, and musketeers using firearms on horseback, using a variety of names.

During the 18th century and early 19th century dragoons became popular, along with Hussars. These were light cavalry who had little in common with the earlier Polish Hussars and were used as scouts and foragers. They primarily used sabres, curved swords. The armoured cavalry of earlier periods were replaced with Curaisseers, metal armour covering the torso and a metal helmet. Some nations preferred simple heavy cavalry, which was a man with a large horse and sword.

This continues in Europe until the late 19th century, where my knowledge thins. Although the Allied plan for the first battle of the Somme relied on German cavalry not being there to disrupt it.

Main source: http://warfare.uphero.com/Renaissance/RenaissanceWarfare-AirfixMagazineArticles.htm