Tactics where they march in orderly line instead of charging, making themselves easy targets, not using cover, cannonfodder.

The accuracy of black powder muskets was incredibly poor; beyond 50 yards most fire would be considered unaimed (and British training manuals in the early 18th century did recommend aimed fire), and beyond 150 yards the chances of a bullet hitting anything were quite small. Keeping men in compact units allowed for this inaccurate fire, that would otherwise be of little effect, to be concentrated. Considering also that the smoke produce by the powder of the muskets and cannons could leave a heavy pall over the battlefield, this further militated towards compact units. Not only would fire in these conditions be inaccurate, but officers would need close control over their men, and easy formations to maneuver with.

Why were these tactics used? What was the thought process behind it? What was the mindset behind them and what brought the change into modern military tactics we see today?

I believe I've explained the first two parts. It should be added, and this relates to the third part, that skirmish formations and light infantry were used, often in tandem with line and column formations. Jaegers, Voltigeurs, and Riflemen were all utilized in the Napoleonic Wars, delivering aimed musket fire, or more accurate rifle fire, from dispersed formations in advance of the main units. They would both screen against enemy light infantry, try to disrupt enemy formations, and also provide information on the ground that lay ahead, so that units advancing into battle would be aware of any irregularities such as ditches, ravines, knolls, etc.

Starting in the 1850s, such light infantry tactics were increasingly seen as the only viable tactics on a battlefield that would now see the use of rifled muskets and cannons. Less so in the 1860s, but even more so in the 1870s, when better breach-loading rifles like the Chassepot and breach-loading artillery appeared, skirmish lines were increasingly favoured by the German Army in particular, and then by the rest of the Continent, at least in theory. The issues for command and control and discipline which this change in tactics posed, were just as if not more important than the quandaries presented by magazine rifles, breach-loading quick-firing artillery and machine guns in the 1890s and early 1900s. Establishing 'fire superiority' was the answer to the latter, though few could have guessed the prodigious amount and types of artillery and automatic weapons this would require. The former was more difficult; devolved command, esp. mission tactics or Auftragstaktik were the solution in theory, but practice was far more difficult, as even practical battlefield communications were still comparatively primitive in 1914.

As for sources, After Clausewitz by Antulio Echevarria is an excellent read for tactical debates, and Gervase Phillips' article "Military Morality Transformed" helps to frame the issues confronting armies with regards to C&C and discipline. I'm less knowledgeable on the early 19th century, blackpowder side of things, but the FAQs section should yield more answers, and more sources. Hope this helps!

To expand on u/DuxBelisarius' answer to the first part of your question: the tactics of 'line infantry' were, by the beginning of the 19th century, and particularly during the Napoleonic Wars, actually relatively fluid, particularly those used by the French army under Napoleon.

Formations used ranged from a 'column of divisions' or 'column of companies', whereby pairs of a battalion's companies would 'stack up' behind one another to form a squat rectangular formation which could be held together easily over rough terrain and could thus move fairly fast; the classic line (used to maximise firepower); the skirmish line a spread-out formation that would range ahead of a unit to drive away enemy skirmishers and harass enemy units; and the hollow square (used to stand off cavalry).

Each had its own purpose, as I've just explained, and no formation could deal with every battlefield situation: a column of companies could move fast and deploy quickly into any of the other formations, but it had nowhere near the firepower of a line; a line had firepower, but was slow and vulnerable to being flanked, especially by cavalry; a skirmish line was incredibly vulnerable to cavalry if it was on open terrain, and could even be pushed away by a determined infantry charge; the square was essentially unbreakable by cavalry, but it required a constant hedge of bayonets to keep horsemen away, meaning it was vulnerable to enemy infantry and artillery who could break the formation (it was also very difficult to move fast in square for the same reason). Crack troops were capable of going from one formation to another very fast while under fire - which was necessary, since they were most vulnerable while deploying from one formation to another. Chandler cites the platonic ideal of this flexibility as being the attack of Morand's division at the battle of Auerstadt, which went through five different formations in quick succession as its situation changed, without letting up pressure on its Prussian opponents.

Trained skirmishers not only had to be able to fight 'en tirailleur' (as a sharpshooter), but also had to be able to recognise dangerous situations and respond very quickly to commands from officers, relayed either by drum or cornet. The light company of one of the British battalions of Guards lost seventy men and nineteen officers killed or captured at Fuentes d'Onoro when it failed to form a square quickly when surprised by French cavalry. A tighter formation allowed an easier move to square, and allowed officers to keep tighter control of their men, with the battalion's commander knowing that every one of their soldiers could hear the unit's drummer.

Standing off cavalry required a unit to be formed up. A hollow square (the ideal formation for infantry defending against cavalry) was a tight formation whereby each soldier used their bayonet to form a 'wall' which enemy horses would reufse to charge. The unit could then drive cavalry off with firepower, safe in the knowledge that they were not in danger from cavalry. Before the advent of rifled muskets, there was no real way for an infantry unit to stop cavalry with firepower alone (i.e. without the square); the range of a musket was just too short and the rate of fire too slow for a unit to break a cavalry unit before it had closed the distance. Therefore, units had to be able to change formation quickly to prevent themselves form being decimated.

Further, formed units were much easier to use to charge. Generally, an infantry firefight would, if one side were not completely broken by fire, be decided by the bayonet. A skirmish line was very difficult to press a charge home with, but a tighter formation could be used to charge. Usually (unless the enemy were defending a strong-point like a redoubt or a village) such a charge would not result in melee combat - either the defenders would retreat when they saw the enemy charging, or disciplined fire would break the chargers. Keeping a charge together or defending against a charge depended a lot on keeping men's courage and discipline together - much easier in a tighter formation where they could be cheered on by their comrades and spurred on by officers than in a skirmish-line, where each man had to assess the situation for himself.

Sources:

The Campaigns of Napoleon by David Chandler

Swords Around a Throne: Napoleon's Grande Armée by John R. Elting

Redcoat: The British Soldier in the Age of Horse and Musket by Richard Holmes

Rifles: Six years with Wellington's legendary sharpshooters by Mark Urban