Smash is the largest icon on the main menu for a reason. This is the most basic way to play. You can set the rules, play alone against bots of varying skill levels, or against friends, locally. You can even watch computer players fight each other.
This latest game has fewer modes than previous games. Coin and Stamina battles are unavailable, for some reason. You can choose between Timed or Stock matches, with anywhere between 1 and 99 minutes/lives. You can also set the time limit to infinity, but then, of course, the game will never actually end without quitting to the menu. Free-for-all and team battles are available, and you can set the teams however you want.
More advanced options include setting damage handicaps, ease of launch, which items appear (though not how often they appear, for some reason), and which stages will show up when you select "Random."
After playing for a while, you'll unlock additional rules, including time limits for stock matches and adjusting the score penalty for self destructing in Timed matches. It's not clear exactly how these rules are unlocked, but it seems to be after scoring a total of 200 KO's in Smash mode.
Online Smash battles with friends are available from the Online menu and work the same way with the added bonus of lag.