Should I use the adapter or the battery?

I want to know when my MBP's battery was fully charged it's better to unplug my adapter and use my battery power or still let use adapter power?

I somewhere read that when MBP's battery was fully charged it's just use adapter for power and don't use battery, if it's true it didn't hurt adapter and reduce it's lifetime?

I have Macbook Pro 13" dual core(2.53)

Read what Apple has to say about notebook batteries:

1. Apple’s Batteries Page
2. Apple’s Notebook Batteries Page

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

1. lithium-ion batteries need to be fully discharged and recharged at least once a month. Go ahead and create an iCal reminder…
2. Leaving it plugged won’t affect it, as long as you remember to do (1) every 30 days.
3. 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.
4. 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.

I think the rules regarding battery management are related to the model MBP you have. The new unibody MBPs have much better battery management, and those batteries don't fall for the same ills that plagued the previous pre-unibody models.

I've had a unibody MBP at work since they were first released, I keep it plugged in all day, every day, and the battery has been fine when I go my meetings throughout the week. I have a pre-unibody MBP at home and I've burned through two batteries in less than four years.