Tea Infusers are tea ware that help you brew tea leaves in a container, which acts something as a teabag. Tea infusers will contain your loose tea leaves and prevents bits of tea leaves (most of it anyway) from mixing into the teapot or kettle.
In most cases, it makes easier cleaning of the teapot when you are trying to remove and a dispose the tea leaves after a tea session. The big teapot at pix 4 gave me a hard time when I try to remove the tea leaves when I am washing up. I would normally fill the pot with some water and turn it upside down to wash the leaves out.....which may be take an additional rinse or two most of the time.
Should you get a tea infuser? No....its not necessary. But if your teapot comes with one as in pix 1 and 2, then it might be a fun experiment to try brewing your tea with the infuser.
But for the bigger tea pot in pix 4, I would now brew my tea in a smaller teapot and pour all 10-12 infusions into the bigger teapot.
The stainless steel tea infuser in pix 3, is about the size of a baseball. This can be opened by unscrewing the infuser and placing your tea leaves inside. This infuser comes with a chain and you can use it like a giant tea bag and jiggle it in a teapot of hot water.
There are now many fanciful infusers available for you to use. I have seen the Star Wars Deathstar model and even a Jaws shark infuser. These are eye catching and may even be a collectible hobby for the tea enthusiast.