Eventually tires wear out, and need replaced. Typically the rear tires start to square up, losing their rounded profile, as the center of the tread wears away faster than the shoulders. Front tires generally wear more evenly across their tread, but may begin to develop scalloped wear known as cupping. Knobby tires are more obvious as the knobs start to wear, tear or break-off over time.
Inspect your tires for adequate tread depth. When the tire is worn to the built-in indicators at 1/32nd inch (0.8 millimeters) or less tread groove depth, or the tire cord or fabric is exposed, the tire is dangerously worn and must be replaced immediately. Also inspect tires for uneven wear. Wear on one side of the tread, or flat spots in the tread may indicate a problem with the tire or bike. Consult your local dealer or mechanic for help. Inspect your rims also. If you have a bent or cracked rim, it must be replaced.
A good practice is to plan ahead and have replacement tires lined up and ready to install before the old ones are totally worn out. Tubes should be replaced at the same time as the tires, on tube types. Old tubes deteriorate and are prone to cracking, which can lead to sudden failure, so a new tube should be installed whenever the tire is replaced. Make sure the tube (if it is used) is the right size and is compatible with radials if need be. Rim strips should also be replaced if they look deteriorated.
On tubeless tires, it's also a good practice to replace the valve assemblies, as the rubber deteriorates. Tire pressure monitoring system sending units in the wheels of some high-end, modern bikes should also be checked and their batteries replaced as needed.