Possibly. It all depends on the frame.
For a multispeed bike the frame will need:
1. Guides for the cables controlling the front and rear derailleurs.
2. Fixing point for the rear derailleur "hanger" (the front can just "clamp" around the seat tube), if you're looking for a derailleur-style bike.
3. The right kind of dropouts for the rear wheel.
A dedicated single speed probably won't have either of the first two and may have the wrong kind of dropouts.
In addition you'll need shifters, brake levers, derailleurs, new rear wheel, possibly new brakes, new crankset...
Bottom line...whilst it might be possible to do a conversion, for a 20 or 30 speed bike (one with front and rear derailleurs) you're better off buying a new bike.
An alternative might be to get an internally-geared rear hub. Shimano and others make them; "cruiser" style bikes often use them. You'll have far fewer bits to upgrade, but the frame may still be an issue.
I recommend you take your bike to your local bike shop. They can take a look at your current ride and help you determine if conversion to a multispeed is possible and, if so, at what cost.