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.
You can choose timed, stock, or coin matches (where damaged players drop coins, and whoever has the most coins at the end is the winner), 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.
Special Smash rules are also available, allowing you to play stamina matches (where a HP meter drains to 0, rather than a damage percentage that rises) and add other effects, such as permanently equipped Rocket Belts, huge characters, increased speed, and other things.
New to this Wii U version, up to 8 players can play locally, in free-for-all matches or in up to four teams. Only specific stages are available in 8-player Smash, and only stock and timed matches are playable.
More advanced options include setting damage handicaps, ease of launch, which items appear and how often they appear, and which stages will show up when you select "Random."
Online Smash battles with friends are available from the Online menu and work the same way with the added bonus of lag.