Think of it as a sweater for your teapot. A tea cozy is a wonderful item that really does help to keep the tea in your pot much warmer than it would be in an uncovered pot. Tea cozies come in a variety of styles, sizes, and fabrics so that there is a style and color to go with just about any tea set.
Some tea cozies are designed to slip over the entire pot, like this red Chinese style as well as the black embroidered cozy in the top picture. This style covers everything and is removed when you wish to pour a cup of tea.
The other common style is like the pink and white cozy shown above. It has two panels with open sides, joining at the bottom in a elastic band, and at the top where it is gathered and tied on with a ribbon. This style remains on the pot while pouring as the spout is uncovered. This was the first cozy I owned and I have enjoyed it for many years. The only problem I have with this style is that often teapot spouts drip and the tea ends up on the cozy itself. (Yes, you can use a Dripper Sipper to avoid this, but I don't usually remember to grab one.)
How should you choose a tea cozy? Decide which style you think you would prefer. Maybe you would like the slip-on model because you would like to show your teapot's designs. Maybe the tie-on would help to unify an unmatched tea set.
I think it is overall, just a matter of personal preference. I would suggest that you make sure the cozy is well-padded, as this adds to its ability to insulate your tea. I would also suggest that you take your teapot with you when shopping for a cozy, as different ones will not fit certain pots. It should be snug, but not tight. My cozies fit most of my teapots, but not all. I have a few teapots that are quite tall and some that are too fat. All three of these cozies fit my everyday teapots just perfectly.
All this talk about tea is making me thirsty. Guess I better go put the kettle on! Won't you come join me?