Your computer's charger and the computer itself will prevent overcharging of the battery. Depending on how your computer's manufacturer set it up, you may actually see it stop charging after 95% or 98%, but most will show it going to 100% and still say charging, even though it is actually running off of the charger but not adding power to the battery.
Even with that, battery wear is an inevitable part of using a battery in any fashion. Modern batteries (Li-Ion, LiPo and NiMH) do not suffer from the "memory effect" that plagued earlier (NiCd) batteries. In fact, most actually function better if recharged more frequently and not allowed to discharge all the way every time.
Still, all batteries have a limited number of recharges before they start wearing out, but you should expect them to stay good for at least a year or two with moderate use. My 17" laptop went for about three years before the battery life dropped below an hour, and I replaced it with an aftermarket battery nearly two years ago that still runs it for over two hours.