Every tire has a maximum inflation recommendation (in Pounds per Square Inch - P.S.I.) labeled on the side of the tire. If you follow this number, you and your tire will be happy campers (or cyclists as the case may be). Riding with your tire pressure too low can lead to 'pinch flats'. Pinch flats happen when you hit a bump with an under inflated tire and the tube gets pinched between rim and tire casing. The tube tears in two spots and looks like a snake bite. Riding your tires too high can make the ride feel bumpy and harsh.
The best way to inflate tires is to buy a good floor pump. Gas station air compressors aren't very accurate, can fill too quickly and don't always deliver high-enough pressures. A good floor pump has a gauge, so there's no guesswork and many come ready to use both types of valve stems (see below). On the road, a small hand pump or carbon dioxide inflator is a good idea so you can change a flat tire
A road bike tire at 100 psi can just barely be compressed when pushing on it with your thumb.
Valves come in two styles:
Schrader and Presta. Schrader valves are the same type as used on your car and many mountain bike wheels. Presta valves are generally found on higher-end bikes. Just remember to buy tubes with the right valve style and you'll be set.
Schrader valve tube (image above)
Presta valve tube (image above)
You'll need to top of your tires every 4 days or so as the butyl used to make inner tubes does leak when pumped up to higher bicycle tire pressures.