A "vertical slice" is a fully functional version of a game with very limited content (usually one level). 

This is very different from a typical "prototype", where basic functionality for the entire game (or much of it) exists, but hasn't been fully developed yet - art might be literally cubes and circles, lots of bugs may exist, etc - the goal of a prototype is both to get the "bones" of a program working to prove viability and provide a scaffold for iterative development.

Think of a vertical slice, then, as a "demo version" of a game - not every feature may be present, there is probably only one level, extra characters might not be unlocked - but everything *works*. I can hand you my vertical slice and you'd get the play experience (albeit limited) that a real user would get with the real product. This includes polish such as sound integration, particle effects, etc.

For the GAME Showcase, we want to emphasize polished, complete (if short) experiences. Games submitted should feel "finished", even if they're more limited. For instance, if you make a 1st person shooter we'd rather see one PC, one enemy and one gun where each of those things works really well, looks good and "feels" good than a game with 50 enemies and 80 guns but nothing is *done*. The same goes for levels - in nine weeks you're not going to make your Gaming Opus, most contest games feature only a few minutes of gameplay but are polished experiences. The way our judging and feedback is set up, a tightly scoped but well polished game will win over something that is broader, more ambitious, but less complete.