Decide what you like and dislike about your old tires. Do they get too many flats? Do they work well off road and on?
Understand that some tire problems may have been caused by not riding with the right tire pressure.
Look on the sidewall for the exact dimensions of your old tire. You'll need this information when you go to the bike shop, but keep in mind that you have the option of going to a wider or thinner tire.
Consider a front-specific or rear-specific tire for a mountain bike if you ride off road. Ask your local shop what they recommend - a tire that works well in one region's terrain may not work well in another.
Consider a slick tire or a tire with an inverted tread for a mountain bike if you rarely go off road. These tires won't buzz on the pavement and will improve your cycling efficiency dramatically.
Consider a wider tire for your road bike if you've been getting lots of flats or if your bike is harsh-riding. Many road bikes are sold with 700-by-20 or 700-by-23 tires, but 700-by-25 or 700-by-28 tires offer better comfort, better durability and fewer flats.
Consider kevlar-belted tires if you get lots of flats. You might also consider special tubes. Ask your local shop what they recommend.
Consider folding tires if you want to save weight. Realize that these tires often cost twice as much as the non-folding equivalent.