Super Smash Bros. is a series of platform fighting games. It can be thought of as “king of the hill”, where you and your opponent try not to fall off the hill (the stage), as doing so results in the loss of a life (or stock) when the boundaries of the stage (blast lines, which is a large box surrounding the stage) are reached. To make your opponent reach the blast lines, you attack them. Attacks deal damage in the form of percent and knocks them back. Percent is unlike health in traditional fighters in that lower percent is more desirable, as having more percent increases the distance you get knocked back by, increasing the chance that the blast lines will be crossed, resulting in the loss of your stock. Upon the loss of your stock, your percent resets back to zero. Once the number of stocks of a player reaches zero, the other player wins.
Here, Peach has 3 stocks and 0% while Marth has 4 stocks and 128%. Peach has less lives than Marth so she's losing, but Marth will likely lose his stock soon because his percent is too high.
Competitive Melee (the main competitive game of the Super Smash Bros. series) is played with a timer of eight minutes and four stocks each, often in best of three matches (best of five for grand finals, top eights, or top thirty-twos in some cases). It can be played one-on-one (singles), or two-on-two (doubles) with friendly fire on. Only some stages are permitted for play as others can have random elements, stage hazards, or abusable properties (ground leading into blast lines (walkoffs), large enough for stalling to occur).
In the above gallery is all of the legal stages in Melee with a red box representing the blast zone.
In most cases, you'll be using a GameCube Controller (unless you're playing the original Super Smash Bros. for the Nintendo 64).
*Can only be used in doubles when your partner has more than one stock and you have zero stocks. Will otherwise pause the game.
While it might seem that there's a lot you need to learn (which is very true), there are only a couple important things you need to know: the control stick is to move, Y and X (and tapping up on the control stick) are for jumping, L and R are for shielding or dodging, and A, B, and C-stick are for attacking. To perform different attacks, hold a different direction on the control stick. Everything else is used for more advanced play.