I used to be lead beta-tester for a now-defunct game company (in part because they didn't listen to we testers!) so I can answer that!
You should have to sign an NDA - a non-disclosure agreement. That limits you to only publicly discussing generalities about the game. Usually they'll then send you a CD (minimal, if any, docs) or point you to a secure download site. The lead developer usually will choose testers from a broad mix of backgrounds who'll bring different perspectives to the game; from average gamers to hardcore grognards - the target market.
Then you TEST the game. It's not just playing it, it's trying EVERYTHING you can think of within the game. Make sure it does what it's supposed to do properly, but also see what happens when you do stuff you're not supposed to be able to. You've got to put the game through it's paces, not just play what you want. It's not really all that much fun - you're testing it, not enjoying it. And most beta code will crash your sytem a few dozen times

. Be prepared to get thoroughly sick of the game!
Keep good notes about what heppens when you do X, the more detailed the better.
When something untoward happens, is it recreatable or transient? Is the UI too cumbersome or too sparse? Are the game events happening correctly or is the Iroquios chief being elected HRE?
Usually you'll be on a tester-only email list or webboard and provide feedback that way. Every company does it differently, but usually all the feedback will be compiled and prioritized from "Fix Now!" gamebreakers to "If we have time" enhancements. Then the code warriors do their thing and release another version and the cycle repeats.
Standard reward is a free copy of the retail version of the game and a mention in the credits. As a beta tester, you have no liability

.
Now...on the issue of bugs. EU is actually pretty bug-free, comparitively speaking. What most folks don't realize is that the codejockeys don't work in a vacuum - they've got budgets and deadlines to work to. When a release date is selected, the marketing folks get busy and start lining up distributors (who get very irate when your shipping date slips!), advance advertising, etc.; printers get contracts for the boxes and manuals; duplications houses get contracts for the CD burns; and so on. These deadlines can usually slip a little, but not much which means that often the programmers are stuck between rocks and hard places; fix Fiji's port problem or fix over-rampaging natives? Sometimes you just HAVE to ship knowing that there are bugs because there isn't time to fix them all.
Given the complexity of the code (and it's actually harder to program for openness like EU has for user mods!), it's actually surprising that the bugs are all relatively minor - no gamebreakers!