I've made a number of small bowls - sometimes called pill bowls or pinch bowls.
To address Gary's question, I use a glue block, and a standard scroll chuck. I have several glue blocks that are threaded to screw onto my lathe spindle, but I could just as easily grip a glue block in my scroll chuck. I start by attaching the blank to the glue block with double-stick tape (hot-melt glue would also work). I turn the blank to round, and cut a mounting tenon on the tailstock side of the blank. The end with the tenon will be the bottom of the bowl, and I always leave the tenon as a foot, so I make sure that it is sized for the optimum diameter of my chuck jaws so that the jaws don't mar the outside of the tenon), and I create a shallow recess inside the tenon so that the finished bowl sits only the outer edge of the tenon without rocking. Sand the bottom (possibly adding some rings for decoration), and then reverse into a scroll chuck to turn the top and inside. I usually use an oil finish - either a commercial Danish oil product, or a shop-made equivalent.
As to tools, all you really need is a spindle gouge. A typical set of carbide tools would also work.. Anything beyond that would be for convenience or perhaps specialized decoration.