The teapot you choose depends very much on the tea you are drinking. What you use for a fresh green, oolong or white tea is different from that of an aged pu erh or aged oolong. For a fresh tea, I personally prefer a ceramic teapot, and I'm particularly partial to a gaiwan (three pieces: saucer, cup and lid). The ceramic doesn't retain the tea essences over time, but I find it allows the tea to more fully open both in flavour and aroma than a glass teapot which could also be used. I find the thinner the ceramic the nicer the final beverage (my favourite teacups are nearly paper-thin and almost translucent). That said, with finer ceramic the heat transmits very quickly making it difficult on the fingertips of some. The gaiwan takes practice to learn to pour from as well, but with time you'll burn your fingers far less!
For an aged oolong or pu erh I would use an unfinished fired clay teapot. These are particularly special. Depending on your own special interests, some people reserve a single teapot of this kind to drink only with a specific tea (i.e. a particularly special aged sheng pu erh). Personally I reserve one for my oolongs and one for my pu erhs, though I have my preferences as well. Part of the reason for using the unglazed teapot for these teas is that you actually want them to retain the tea's essences over time because those essences contribute to the flavour of future pours, giving future teas an even richer, fuller flavour.
No two teapots are the same or created equal either. Hand-thrown pots are far superior to slip-cast teapots and the clay of Yixing in China is superior to almost any other; it is certainly the best known in Chinese tea drinking circles. Because the clay is so thick and porous it better retains the tea essences faster than clay for finished teapots from other places. In addition, the artisans in Yixing produce masterful work. These teapots are easily distinguishable from others. That said, the artisanship in places like Yinge in Taiwan cannot be discounted either: but the clay is not AS good for the final product. However, there have been some real teapot innovations coming out of Taiwan such as the bottom pour! One of the reasons for selecting a hand-crafted teapot are its chi properties, which are far superior to a slipcast model. As your teapot retains tea essences, it melds the chi of the teas held and poured from it, growing its own chi further.
When selecting your teapot you also don't want one too big or too small for the number of people you are serving. With practice you'll know the size that is right. I find it's nice to have a 1-2 person teapot, 3-4 person teapot and 5-6 person teapot. Any more people than that and you can double steep in the gong fu tradition pouring from the tea pitcher only every two times you pour from the teapot. The 1-2 person teapot is perfect for enjoying tea by yourself and is also nice to reach a degree of meditation you may not otherwise experience with a larger group of people to distract. That said, sharing tea is an age-old tradition, particularly in China and drinking it in this fashion is perhaps unique for many outside of Asia.