I never really looked at one till I was working on my CFI, and that was only a cursory glance. They never came up in any checkride. I've since learned a good deal about aircraft maintence (most of it not at any school).
A maintence logbook is simply a chronological record of all the maintence and inspections performed on a airplane. It can be as simple as a spiral notebook, or as complex as the monsters carried on the airliners.
Many airplanes have seperate logbooks for the engine, propeler & avionics. This way the logbook for the engine can be transfered with the engine if it is ever sold. This also keeps the master A/C log from getting too complex.
99% of the time you will only care about the A/C log. It is the master that will stay with the plane from the date it is built till it is scraped.
Anytime you are flying a plane you have the right to see the logbooks for that plane. Most of the time if you ask to see them, the owner won't let you take them out of his sight. This isn't usually an issue with the maintence, but they are worried about you loosing the book. A plane with lost logbooks looses about 20% of it's value. Anytime a FBO refuses to let you see the maintence logs, take your buisness elsewhere.
As the pilot you are responsible for making sure your airplane is in an airworthy cndition. You aren't expected to be a expert, but read the planes history to see if there is anything recent that you need to be aware of. Many times a problem will occur due to recent maintence. For example I would be very concerned about flying a newly installed engine over water, but if it's been running fine for 1000 hrs I wouldn't worry as much.
The things you as a pilot need to look for are:
The last annual and/or 100hr inspection.
Pitot/static check in the past 24 months (useually a part of the annual)
Transponder check in the last 24 months (also usually a part of the annual)
ELT battery check/replacement.
ADs should be complied with as part of the annual/100hr.
Most of the time VOR 30 day checks are kept seperate from the logbook, since they are done so frequently. Most owners I have seen keep a small notepad in the cockpit for this.
As a practical matter this is all you will be expected to be familiar with as a pilot.
Pilots and owners can performe "preventative maintence" which is defined in FAR 43, apendix A, para c (it's a long list that will surprise you). If you chage a tire as allowed under preventative maint. You will have to make a logbook entry that includes the date, A/C time, a description of the work performed, your name, and certificate type and number.
Example:
2/21/05 3498 TTAF
Removed wheel and tire. Installed new Goodyear tire per maufactuers instructions. Reinstalled wheel per Cessna service maunal 12-54-B. Performed taxi and brake checks.
Scott Beadle PPL# 1234567