Class and Level System

When Character Class System meets Character Level. The oldest, and arguably the most popular, type of Game System. A player chooses a class at character creation, and as the game is played, the character will earn Experience Points; when they earn enough, they will advance to the next Character Level, which will grant them new abilities and improve old ones (usually).

When multiclassing is allowed, it will probably be per level; each time the player levels up, they choose which class gains the level. Think of a description such as "My character is a level 3 warrior, level 1 thief, and level 2 necromancer". Sometimes a character can use the experience they gain from their basic class to advance to a more powerful, specialised version of that class. The older games also featured monsters and abilities that could take levels away from characters, often forcing them to gain those levels back the hard way.

The Job System is a specific version of the Class and Level System, where classes level independently of each other; each class is like a different character, and the character can switch between them at will.

Examples of Class and Level System include:

Hack and Slash

In Diablo and its sequel, you select one of several different character classes, but how you develop the character is up to you. In the first game, leveling up gives you five stat points you can add to your strength, dexterity, life or magic however you see fit. In Diablo 2, you also get one skill point with each level, and can add it to any accessible skill on one of your skill trees.

MMORPGs

World of Warcraft uses this. There are point buy systems (talents) but these are exclusive to each class.
City of Heroes has a fairly traditional Class and Level System, in that it has classes and levels, though the classes themselves are more exotic than just the standard warrior, mage and cleric. Unusually for a MMORPG, it also lacks a Point Buy System entirely, instead offering new power (skill) choices on some levels, and slots for enhancements on others. It even avoids the traditional act of taking the same skill multiple time to get better versions of it, relying instead on the enhancement system for skill improvement.
Wizard 101 follows this formula, although it uses a more simplified version, for the benefit of its younger gamers.

Roguelike

Angband and its many variants use this. The original has no class customization at all, but some variants have added skill systems with point buy.
ADOM
Nethack
Dungeon Crawl
Tales of Maj'Eyal offers heavy class customization based on talents. Each of the many classes has a large amount of buyable talents, and the classes themselves are very different in talent selection. Most races also offer unique talents.

Role-Playing Game

The Elder Scrolls games tend to be hard at the onset no matter what you do because of your low (arguably fair) chance to succeed at anything you do, and it is insisted that you train in a combat skill regardless of what your build is. Older The Elder Scrolls games have guides explaining the way stats work and the peculiar hoops you have to jump through to maximize them properly.
- Skyrim does away with traditional classes entirely.
Similarly to the Diablo example above, in Demon's Souls the class you choose only affects which items you start with and your initial stats, but from that moment onwards, you can increase whatever stats you wish and it is very possible for a mage to end up wielding a Dragon Bone Smasher (a gigantic sword).
Dragon Age Origins has the typical classes. Leveling up gives you three points to spend on attributes to increase, can spend one point to learn a talent/spell, etc. And Specializations act like Prestige Classes.
Mass Effect plays with this. The Class and Level System is in full force, especially in the first game. However, almost every character has their own unique class. Only Commander Shepard has a choice of classes, and only Kaidan and Ashley have classes that come from the same pool (and even then, only in the first game).
Final Fantasy was one of the first eastern Role-Playing Games to use something like this. While the first one just let you build a four man party out of six classes, later games such as III, V, X-2 and Tactics let your characters freely change to any class you want and even mix and match abilities between them for customization.
Mostly played straight in The Last Remnant; your character's class is based off the skills they use, their level in that class is then based off their stats. For example, mostly using combat arts may give a character the Gladiator class; if they also had 57 strength, they would be an Adept Gladiator.

Tabletop Games

The original Dungeons & Dragons is the archetypal example of this kind of system. Third edition added some point-build aspects (feats), as well as a very flexible multi-classing system. Fourth edition gave every class a fairly large palette of abilities for the player to choose from, but those abilities are almost always unique to that class.
- Hackmaster seems to be for people who think 3rd Edition Dungeons & Dragons is for effete metrosexuals who wouldn't know which end of a dagger to hold without a self-help book. Thus, the 1st/2nd edition Dungeons & Dragons system is repeated.
The similarly vintage Traveller science-fiction role playing game had classes (Careers), but not class-levels in the classic style, opting instead for various Skill Levels (Pilot 1-3, Handguns 1-3 etc). Famous too for a character-generation system that forces players to make a tradeoff between being 18 (full stats, no skills) and, say, 54 (many skills, but stats reduced due to aging). Just to keep the pressure on, there is a significant chance that your character will die during generation.
The Palladium system, used in Rifts and all other games published by Palladium Press, is a pure Class and Level System.
Iron Crown Enterprises, famous for an early Middle Earth RPG as well as the ~~Rollmaster~~ Rolemaster series, did this in most of their offerings.
Monte Cook's World of Darkness is something of a bridge between the classless Point Build System Old World of Darkness}} and the Class And Level themed DnD or D20 system. While you can only be one type of supernatural, and you gain levels, you can choose a major and a minor focus, such as fighting, stealth, or intellect, which affect what skill point breaks and bonuses you get. This focus can change with no penalty every time a character levels.

Turn-Based Strategy

In Disgaea and other Nippon Ichi SRPGs, the prerequisites tend to be more varied, but the classes still dictate the stat growth and equip percentage.

Third-Person Shooter

Transformers: War for Cybertron has a multiplayer character system that's equal parts this and Modern Warfare's "create-a-class". The weapons and abilities are divided among four classes: Soldier (Warrior), Scout (Thief), Scientist (Wizard), and Leader (which has elements of Warrior and, to a small extent, Wizard). Within each class, you can choose any two weapons, two abilities, and three upgrades available to that class (plus aesthetic elements like body style). Each class levels up individually to a maximum of 25 per class, and leveling up unlocks additional abilities and upgrades for that class.