If I keep the power cord plugged in all of the time, will it damage my laptop?
No, quite the opposite. LiIon batteries (which all modern laptops use) wear out from being discharged, so using mains power when possible is advisable.
For more details see e.g. http://superuser.com/questions/12358/is-it-better-to-use-laptop-on-battery-or-on-ac-power
Actually, the best thing for the battery is to charge it to ~%40, take it out and store it in a cool (not freezing) place, as it deteriorates quicker when it's warm. However, that may not be practical, because
a) some laptops don't work well without their battery and
b) This means you will have to remember to charge your battery before you can actually use its full capacity; this might be annoying if you often use it on the road.
In that case, best thing is to always keep the laptop plugged in, to avoid cycling the battery.
Actually, the best thing is to periodically plug and unplug the battery so it swings between 80-90% charge, but again this is probably not practical, as it has to be done manually.
If you keep it plugged in, the battery will not have a chance to discharge. This will cause increased temperature and chemical reactions will begin to shorten the life of the battery. This will affect the battery length in the long term, but batteries can be replaced.
It will not permanently damage any components other than the battery.
Basically if you want to keep it plugged in all of the time, remove the battery at about 40% charge and store it in a cool dry place.