While it would be possible to go through and list every unit in Shogun 2: Total War, it wouldn't be very efficient. That is because units in the game are essentially broken up into certain "types" based on their weapons of choice.
Bow - Bow equipped units are the most basic ranged units in Shogun 2: Total War. They're able to rain a deadly metal hail against opponents, and they're able to easily fire over walls. Bow units are great forcing an opponent to fight a battle on your terms, and they're also great in siege battles. Bow units are equipped with melee weapons, but will lose against similarly trained melee units.
Gun - Units equipped with guns (matchlocks) imported from Europe can do incredible damage to enemy units quickly. However, guns have shorter range, don't reload as quickly, and can't fire over walls and allied units. They're great for defending a fortification or fighting on an open battlefield, but not good for attacking an enemy fortification or fighting in dense terrain.
Naginata - Similar to the Yari, the Naginata a spear-like weapon with a broader blade that is given to some elite troops, like Naginata Warrior Monks and Nagainata Samurai. This weapon is a compromise - it's good at everything, but not amazing at any one task. However, Naginata units tend to be strong because this weapon is only given to highly trained warriors.
Siege - A number of cannon and rocket units exist for siege situations. These units offer great range and the ability to do considerable damage against static units. However, these units tend to be expensive, and they're entirely useless in direct combat.
Sword - Units equipped with swords, such as the Katana and the No-Dachi, are very effective in melee combat against infantry. They tend to be weak against calvary, however, as they often take great losses from a calvary charge. Still, sword units are the best choice for melee slug-fests.
Yari - Yari is the Japanese word for spear. Units equipped with the Yari are particularly good against calvary. Most Yari units have a respectable charge bonus, as well (in other words, they're great for charging in for an attack). Yari units can do well against sword infantry with proper training, but cost-for-cost a sword unit is typically superior in melee.
Now, with this addressed, let's talk about the different levels of training you'll find among these unit types.