Over the years, we've bought mostly Legos because of their quality and variety and number of kits. We have a few Megablocks from Canada, my son is a big fan of Pirates of the Caribbean and MegaBlocks got the contract to make constuction kits for the franchise. A few years ago we spotted some Kids Connection bricks at Wal-Mart, they were extremely cheap and in a neat storage case, so we snatched them up. I've recently discovered another company, Best-Lock out of England.
Legos are to found everywhere. Kids Connection is a brand owned by Wal-Mart, so they're exclusive, but they look a lot like MegaBlocks, so I bet MegaBlocks makes them for Wal-Mart and several other retailers. Mega-Blocks are found everywhere. Toys'R'Us has Best-Lock on their shelves and on their amazon.com portal.
I'm sure there are a dozen different companies that copy Legos or come into their own, these are just the name brands that I've come into contact with.
I just found out that K'NEX also makes good quality bricks that are compatible with Legos, but we don't have any K'NEX bricks, just the sticks and connectors. (thanks westfw!)
Step 1: Lego my construction brick!
They are made of relatively dense plastic and very opaque. They usually have very little flash on their pieces. The denser plastic also makes for better adhesion. If the bricks make a sharp click when they stick together, they're going to be harder to seperate. (Thanks for the reminder, LilPunk32!) Legos almost always click, and some times require the use of the Lego Brick Seperator tool. The tool comes in handy when dealing with flat plates that require leverage.
You can order all sorts of bricks on-line at lego.com through their "Pick-A-Brick" program. You can also order any piece from any set over the phone, but it'll cost ya, they have to ship them from their headquarters in Denmark.