Competitive play is paying attention to EVs, IVs, movesets, following clauses- the works, against other people who do the same. And that usually involves wifi unless you happen to be lucky enough to live in an area with many other people that play competitively as well. Unless you use Shoddy Battle or something.
Pretty much no one goes to tournaments. I know nothing about actual tournaments that are held places. But no one goes to them, so meh.