House of Troy Brass and Black Grand Piano Lamp

Do I have to buy specialty bulbs?

No, House of Troy Lamps use common light bulbs that should be readily available at your local hardware store.  And for LED lamps, you should never have to change a bulb.

What is the difference in types of lighting (incandescent, compact fluorescent, LED) choices?

Regular incandescent and compact fluorescent bulbs produce a remarkably similar light.  The fluorescent use less power (lower wattage), don't get as hot, and last longer than regular bulbs.  The LED bulbs produce a very white light instead of the usual yellow-white light.  LEDs are also very energy efficient and will likely never need changing.