I've been interested in checking out some "DIY" boardgames, ones that you could just print the pieces/sheets/cards/tokens/etc and play.
I'm mostly interested in boardgames that can be played with 4+ people and require little setup or rules introduction, as I'm pretty much being the "gateway drug" for the group (everyone I've shown Blokus so far has loved the game).
I'm ok with knowing about heavy games (stuff that takes an afternoon to finish) to keep an eye on for future reference
The term you're looking for is print and play. There's a very large print and play community out there. Websites dedicated to sharing games. The board game industry is pretty open source as game mechanisms are not copyrightable. Only art and rules can be copyrighted.
Assuming you're in the U.S. (though possibly even if you aren't)...
The cool thing about game mechanics (boardgame, video game, school playground game, etc) is that they're not covered by copyright. And (while this bit might be less true of video game mechanics) they're rarely covered by patents.
So, for the most part, clones require no licensing or anything. You can make a knock-off of Carcassonne or Settlers of Catan or whatever legally, so long as you avoid trademark infringement. (Basically as long as you're careful to make it clear your game isn't by such-and-such company and you don't have any affiliation with them.)
(Also, it's worthwhile to mention that some games are as much or more so "flavor" as/than mechanics. In such cases, while I don't know that there has been that much precent in the court, it's likely the flavor would be considered copyrightable. So maybe if you're copying the BSG boardgame, don't include Cylons. Also, IANAL.)