What kind of lathe would I need if I want to turn pens?
There are several basic groups of wood turning lathes based on size.  There are the pen/mini lathes, the midi lathes and your full sized lathes.  

Pen/mini lathes are the smallest. These are almost always bench top models have a small foot print and smaller swing often 8-10".  Bed lengths vary.

Midi Lathes are larger than mini's and tend to have a 10-12" swing, longer beds and most are bench top models, however some have stands.  They tend to be more powerful, since they can handle larger pieces than the mini's.

Full sized lathes are the largest, tend to be floor models (but not always) and have the largest swings, often starting about 14" and up into the 20's.  

In reality you can turn a pen on any of them.  However, full sized lathes are overkill for size, power and price.  Midi's are reasonable to use especially if you turn things other than small items like pens.  

This leaves the pen/mini's as the likely option.  They are the cheapest of the bunch, allowing more money for buying pen stock, the are the smallest, allowing for a small work space/shop, and they are designed for this type of work, which means it should have smaller error tolerances as well.

As pointed out in the comments, these lathes can also be used with resin stocks and other similar materials.