In my mind, "trooper" is defined by not just being a decent all-around 'Mech like Gurdel said, but also requires numbers. In a Succession Wars era timeframe, the Guillotine is decently fast and agile, decently armored, and fairly well armed; but it's also a Star League-era design that's out of production, making it pretty rare and thus cannot qualify as a trooper. These are the 'Mechs that you literally build your military around. The Vindicator is a good example for the CCAF. It was cheap, easy to mass produce, and reasonably effective, so they built them by the DropShip-load, making it a trooper 'Mech.
The brawler, like he said, is just something that wants to get in close and slug it out. If you think about the origin of the term you'll get the picture: a brawl being a large fist-fight usually in tight quarters, ie a "bar room brawl" like you sometimes see in Westerns. Imagine how a 'Mech would fight at punching range, surrounded by enemies, and with little room to maneuver, and you've got it.
