For me anything where at least two people can compete. As I said earlier, almost any activity can be competitive, even hula-hooping (though it has few elements).
What makes a game more competitive is how many elements there are and how much skill is required for each element to play at a high level. (As a side note, a game usually needs to be somewhat enjoyable at low and mid levels of play for it to be successfully competitive.)
Starcraft has many elements are required very high levels of skill: fast fingers, precision, micro, macro, awareness, memorization, etc
Brawl requires high levels of spacing, and lower levels of tech skill and precision. There really isn't much more than that.
Great explanation, we're going to go with that. However there is a few things you're missing. For one, there is a luck aspect when it comes to competition as well; I will get into that later.
Back to your skill conversation, it depends on what you consider skill. My personal view of skill is the ability to maneuver your character how you want to, as well as being able to outsmart your opponent.
Now melee has a stricter tech-based gameplay. In lighter terms, it requires much more time to learn your character and make your character technically sound.. However, because of this steep wall of technicalities, you find that many players don't think outside of a certain box. You find that Melee has players who rely on the idea of being technical getting them through matches; combos, but no internal thought behind it. This is why some (Reflex is a good example) Brawl players are put off by Melee, not because the game is technical, this they can learn in time, however when playing an opponent who you are smarter than, however being beat because their character has gimmicks that you have to technically be prepared for, is discouraging. But I digress, Melee requires a sense of technical power and reaction timing for a large quantity of competitive gameplay, only at high-level do smarts seem to come into the equation.
Brawl doesn't have that technical standpoint. Better yet, it does, however the hardest things that can be done in Brawl, can be done by everyone with pretty minimal time invested in the game. What happens because of this, is that players have to rely on being intellectually better than their opponent in order to win most match-ups. Back to the Reflex example, he used Bowser in tournament (melee wise) and played to outsmart his opponents, however he disliked melee for the idea that he couldn't get past characters like Fox or Falco, not because the player is smarter, but because the player is more technically sound. After Brawl came out, he stuck with it because it suited his intellectual rounded playstyle. He uses Pokemon Trainer (was originally low tier) and got him boosted up just showing what being smart with the character can do. Because of the fact that there are no combos, it requires you to be constantly alert of what is going on with your opponent, reading your opponents actions, etc.
Now, back to the Luck based part of competition, Brawl and Melee have the same Luck-Based qualities, Yoshi's island and it's shy guys, Randall the cloud, G&W's judgment hammer, Peach's bomb plucking, it's all entirely based off of luck. However Brawl DOES have one quality that makes it a luckier game than Melee: tripping.
This is the ONLY thing that would make Brawl by definition less competitive than Melee. However, in most tournament settings, many wii setups have brawl with no tripping as a standard, and they won't play unless they have this setting on their wii. Without tripping, it's a matter of what you prefer more than which is better. All in all, both games are even, this doesn't mean all games are even by the same logic, just that Brawl and Melee are such different games, that it's a matter of opinion which game is better.