Non-Competitive Battles are those done anywhere outside of a Competition, and that don't require a Trainer Club Account (which I don't have, solely because of my control freak dad), and are generally pretty loose about rules and stuff, compared to Competitive Battles.
If I were to battle my sister here at home, via Local Wireless Communications, it would be a non-Competitive Battle.
If I were to Battle her over the Internet, it would STILL be a non-Competitive Battle.
If we were to both do Random Matchup Battles on Battle Spot, at the same time, and (by pure chance) wind up Battling each other, it would STILL be a non-Competitive Battle.
Contrary to what Thi says below, this Blog is NOT "misinformation" in any way, shape, or form, and is COMPLETELY accurate in describing the differences between what I call "non-Competitive Battles", and "Competitive Battles".
I was trying to clarify the differences between the Battles that I've done, many of which WERE Online, and the Battles which the other Players here, like you, Luisjo, and Angela had participated in.
I'm not sure what the appropriate term is, "non-Competitive Battles", "Casual Online Battles", or something else, but there's CLEARLY a difference between completely random Online Battles, whether at the Battle Spot, or a Battle some random person from overseas challenges you to (both of which I've done before), and a Battle following VGC or Smogon Rules.
If you (or anyone else, for that matter) could tell me the appropriate term for the kinds of Battles that I do, and have done in the past, I'd really appreciate it.