Couch Co-Op
Couch Co-Op is often called Local Co-Op, and allows two or more players to play on the same television, game console or computer, and couch...or multiple chairs, or couch and floor...or standing if that makes you happy. The screen can either be split to accommodate the players, or as a shared screen with the players on the same point of view.
Examples: Lego Batman 2: DC Super Heroes, Halo Reach, Minecraft: Xbox 360 Edition
Splitscreen
Splitscreen is a specific form of local co-op that will partition the screen in one of a few ways giving each player their own viewpoint. The screen is often split vertically or horizontally depending on the aspect ratio of the game or the developers preference. Some games get a bit more creative and will have partial-screen splits. There's always plenty of argument over which way the screen is split is most effective, the recent StarHawk game utilized the entire screen while still holding a widescreen aspect ratio and filling in the gaps with a radar.
Examples: Left 4 Dead 2, StarHawk, Splinter Cell: Conviction
Combo Co-Op (Local with Online Play)
Combo Co-Op is a newer term we've been using around here. We used to refer to it as "Local with Online Play" but that term just didn't seem catchy enough. Hey - Randy Pitchford liked it, so why not? This style of co-op allows for two or more local players to play with one or more online players. The set up doesn't necessarily mean the screen is split, but rather local players can play with online players at the same time. This set-up is ideal for co-op playing spouses, significant others, local friends, or siblings to play with their long distance friends at the same time.
Examples: Borderlands 2, Ratchet and Clank: All 4 One, Shoot Many Robots