Before we go in depth about the different types of grips, there's one thing you must keep in mind: No matter what type of grips you have, if they get so old that you can't hold on to the club without strangling it, then you won't be able to swing the club properly. Your grips must give you the feeling that you can grip the club lightly without the worry of losing the club. Some grips last longer than others, but as a general rule you should change grips at least twice a year.
Grips come in different materials that will give you a different feel. The three main types are rubber, cord and synthetic. Most people who play less than once a week prefer rubber or synthetic grips because they are not so rough on the hands. Cord grips are much harder and rougher than rubber grips, but they're much better if your hands sweat or if you play in the rain. All three of these grips can be cleaned with a damp towel. For the synthetic grips, make sure not to rub too hard when cleaning as this can remove the top layer and make them slippery. Don't soak your grips to clean them!
The actual diameter of the grip can have a fairly dramatic effect on your hand action during the swing. Larger grips tend to inhibit hand action, which can encourage slicing, and smaller grips tend to increase hand action, which can encourage hooking. Many senior golfers with arthritic hands actually need oversize grips just to play the game, which is perfectly acceptable.
There are many "grip sizers" on the market that do a nice job of measuring the desired grip size for a golfer. As a general rule, the ring finger and middle finger of the left hand should be comfortably touching the pad of the left hand, but not digging in deeply. You'll need to experiment with what grip size is most comfortable for you, but this will give you a good starting point.
Another factor with your grips will be the style of round vs. 'ribbed'. Some grips have a very slight raised area on the underside of the grip to help with positioning of the hands and feel. In my own game, I have this rib on all my clubs except the putter. This allows me to place my hands in exactly the same place every time on the grip without looking. There is no right or wrong answer on this aspect, it's simply a matter of feel. You should try both kinds before making the final decision.
One final word on choosing the right grips for you: Grips are not expensive, so you can certainly take one of your clubs, have it regripped and practice with that club to see if you like the feel. If you don't, you can certainly try another until you get the one that feels right to you!