If you are just thinking about getting started wood turning and not sure about investing a lot of money, I suggest purchasing one of the smaller "mini" lathes like the Jet Mini Lathe . They are quite inexpensive and don't take up a room, and they are not particularly dusty machines ... yes they throw some dust and chips but generally they are not like a table saw, radial arm saw or band saw that can through out volumes of fine dust and you really do need dust collection. With a lathe you can control the amount of dust to a much higher degree.
The thing that I like about the mini lathes is that if you do get intereted in turning and want to move up to a a larger lathe, you will find there is a benefit to keeping your smaller lather as they can be more useful for smaller projects and just for working back and forth between it and your larger lather.
There are a host of projects that can be done on a lathe from making (woodworking) tools to large bowls, urns, pens, plates Christmas decorations, kids toys ... the list is endless. What I also like about turning is if you want, it is easier to collect and use your own wood. You do not have to purchase all the wood you use, if a neighbour decides to cut down an old apple tree there can be a wealth of great wood ... probably for very little cost (if not free) for you. Because you are generally using smaller pieces of wood, you can experiment with other species of wood and really get some great looking pieces.
You will need a good set of turning chisels that you will be able to use on either a large or small lathe. There are many, many accessories and chisel varaitions fo turning, but to start off with you are better get a bacic sent and start turning to see what kinds of projects you like best. If you decide on doing specific projects you can expand your tool base from their.
There are many many projects that can be done and you may want to pick up some books or DVD on truning to help get you started, again they are inexpensive and will get up up an going quickly. Another suggestion is to investigate a local wood turning club in your area. There are hundreds and hundreds of clubs all over the world and I would be surprised if there wasn't one in your area. If you aren't sure ... check with some of the local dealers ... they will be able to put you in touch with a local club.