How to wifi battle on Pokemon XY and ORAS

If you don't know anyone you can always battle random people in your Passersby list. Or you can easily find challengers on forums or whatever, add their Friend Code and become friends on Pokemon.
You can find these features in your Player Search System (PSS) page. In ORAS the PSS is accessible via the PlayNav app of the PokéNav Plus.

Everything you need to know about how to challenge people or send trade requests is explained here.

The process has an extra step for Omega Ruby and Aplha saphire, but other than that it's the same. I'm going to assume you've already got your Wifi set up on your 3DS, and skip over that bit.

If you're going to battle a specific person over Wifi - you can ask on forums and the like, many will oblige

1. On the 3DS home screen tap the Orange smiley icon at the top of the touch screen to open your friends list.
2. Make note of your friend code by looking at your friend card, which should be the first card in the list. give this to anyone you want to battle
3. in the upper right corner press "Register Friend" (or it can look like the friends icon with a +), followed by "internet", then slot in the code they provided.
4. load up your pokemon game and direct your attention to the lower screen
5. Ruby and sapphire only - launch the PlayNav, this contains the same functionality as the lower screen in X and Y
6. Switch to the Player Search System, and tap the blue wifi icon in the upper right.
7. When your friend connects they will appear in the friends section. Tap tehir face, and select battle

If you just want to battle random people on WiFi

on the Player Search System, connect to Wifi, tap the white icon in the top-center, press battle, press "Battle with someone far away" and then keep tapping highlighted faces among the passersby to challenge them and see if any bite.