Len,
In order to summarize what has already been said by the other colleagues above, you must know that to turn a bowl you need firstly to turn the bottom. That means the first thing to do is to attach the upper part of your blank to the lathe. For doing so you have many options like:
1) A screw chuck:
like this one:
http://www.woodturnerscatalog.com/s...Screw_Center_Chuck___screw_center_chuck?Args=)
or this one:
http://www.axminster.co.uk/product-Axminster-Wood-Screw-Chuck-38mm-23660.htm
2) A faceplate:
like this one:
http://www.woodturnerscatalog.com/s...s___Oneway_Faceplate___oneway_faceplate?Args=)
3) A screw attached to a chuck:
like this one:
http://www.woodturnerscatalog.com/s...__2_Profiled_Jaws___oneway_chuck_2_jaws?Args=)
I didn't mention here but you can also turn the bottom between centers.
By turning the bottom part of your bowl, you'll manage its further attachment in order to hollow your bowl, that is to turn the upper part. Here again you have a lot of alternatives depending of the accessories available.
Generally you turn a shoulder which can be part of a foot or under a foot. The other option is a recess, also under or part of a foot.
Either way, you can attach your base with the accessories mentioned above.
With a screw chuck (specially for small pieces) you'll need to turn an intermediate piece of wood in order to either tap (in the case of a recess) or screw (in the case of a shoulder) your bowl to the intermediate piece itself screwed to your screw chuck.
The same is true with a faceplate.
Finally, with a spigot chuck (like Oneway, Vicmarc and the likes) you either compress the spigot (case of shoulder) or expand it (case of a recess). You understand that the spigot chuck alternative is the easiest one but also the more expensive.
A final thing to consider is that, in most cases, you have to clear the base of your bowl once it's finished. And again you'll need another way of attaching your bowl in order to do so. With regular shaped rims this is possible (among other alternatives) with a jumbo jaw attached to a spigot chuck (see Glenn's PSI link above).
Hope that helps and that my poor English doesn't handicap the explanation. If something is not clear feel free to ask further.
