Should I use the adapter or the battery?
Read what Apple has to say about notebook batteries: 


Apple’s Batteries Page
Apple’s Notebook Batteries Page


EDIT for the lazy(es) who don’t want to read the links:


lithium-ion batteries need to be fully discharged and recharged at least once a month. Go ahead and create an iCal reminder…
Leaving it plugged won’t affect it, as long as you remember to do (1) every 30 days.
If you have to “store” and put your battery away for more than six months, you should definitely store the battery with a 50% charge. Reasons are listed in the corresponding link. 
Pay attention to temperatures, they can damage and severely hamper battery operation.


I hope this makes everybody happy. 

Now… as a final note: the new unibody stuff coming from Apple doesn’t have removable batteries, however, the battery design is the same… those are lithium-ion batteries, no matter how fancy they make it look. Macbooks do not have a nuclear core in there, so the same rules apply, except that… well, you can’t remove it for storage, so more than ever, remember the 30 day rule.