It depends on the vehicle application, the size of the tires, how much weight is on the tires, and whether fuel economy is more important to you than a smooth ride.
Listed in the owner's manual or on a decal in the glovebox or door jamb in every vehicle are the recommended inflation pressures from the vehicle manufacturer. For most passenger cars, minivans and minipickups, the recommendations range from 27 to 32 psi. For fullsize pickup trucks and sport utility vehicles, the recommended inflation pressures tend to be about 5 to 8 psi higher to reflect the larger tire sizes and greater weight of these vehicles.
It's important to note that the recommended inflation pressures may differ for the front and rear tires.
The manufacturer's recommendations are not necessarily the optimum inflation pressure for your tires, but are generally the best for all-round driving. Adding a couple of extra pounds of pressure will decrease the rolling resistance of the tires and make a slight improvement in fuel economy -- but it will also make the tires harder which in turn may cause a somewhat rougher or harsher ride.
If you're carrying a lot of extra cargo, car pooling, hauling a lot of stuff in the back of a pickup or towing a trailer, a few extra pounds of pressure would be recommended to offset the added weight. Add the extra pounds to the rear tires.
WARNING: Never exceed the maximum inflation pressure specified on the sidewall of the tire. This number is the maximum pressure the tire is designed to safely handle. Higher pressure increases the risk of tire damage (when hitting a bump) or tire failure.
All tires leak a little air over time, with some losing up to half a pound a month. If you're losing more air than this, you probably have a leak (possibly a rim leak or a porosity leak in an alloy wheel). For this reason, tire pressure should be checked at least once a month -- and certainly before taking a long trip or driving at sustained highway speeds.
Underinflated or overinflated tires can wear unevenly. Underinflation also increases tread wear dramatically.
WARNING: Underinflation may also increase the risk of tire failure or a blowout. When a tire with too little air in it (say 12 to 18 lbs.) is driven at highway speeds, the sidewalls are forced to flex excessively. This builds up a lot of heat in the tire which may cause it to fail.
Recommended tire inflation pressures are always for COLD tires, which means you should check the tires in the morning before the vehicle has been driven. Driving heats up the tires and causes the air inside to expand. If you check the tires right after driving, therefore, the readings will be at least several pounds higher than normal.
Internal tire pressure will also vary with the ambient (outside) air temperature. Hot weather raises air pressure inside the tires, while cold weather lowers it. So air may have to be added or vented from the tire to compensate for seasonal variations as well.
Use an accurate tire gauge to check your tires. Don't rely on the built-in gauge on a gas station air hose or compressor (which tend to be very inaccurate). And never rely on your eyeballs alone to "judge" the amount of pressure in your tires. The sidewalls on radial tires typically bulge quite a bit even when the tire is properly inflated. If you keep adding air until the bulge is gone, the tire will be seriously overinflated. Likewise, don't wait until the tire is nearly flat to add air. It's nearly impossible to tell the difference between a tire that has 10 lbs. of air from one that has 20 lbs. of air. Use a gauge to check the tires regularly, and add or vent air as needed to keep the pressure within a couple of pounds of the amount recommended by the vehicle manufacturer.
NOTE: Tire inflation pressure should be more or less equal side-to-side. A difference of more than a couple of pounds may be enough to cause a noticeable steering or brake pull.