Clash is a small mini-game tied to certain spells and abilities. When you clash, you reveal the top card of your library, and each opponent you're clashing with does the same. Then each player decides whether to keep that card on top or put it on the bottom. You "win" the clash if your card's mana value is higher than every clashing opponent's card—triggering a bonus on the spell.
The big nuance: clashing isn't about drawing or discarding—the card stays in your library whether you keep it on top or move it to the bottom. Also, in multiplayer you clash against all opponents, and you only win if your mana value beats all of them, which is harder.
In Commander, clash mostly functions as light deck manipulation—setting up your next draw—with the win-bonus being secondary, since beating multiple opponents is unreliable.