A new Twitter client showed up in the news last week and I thought now would be a good time to collect all of the Kindle Twitter clients into one post. (But then I discovered that the new client asn't new; I'd covered months ago.)
So it turns out that the 4 clients I know of are actually 3 clients. As more are announced I'll add them here.
As for using these clients, you'll of course need a Twitter account. Then you'll need to visit the sites from your Kindle's web browser and login using your Twitter account info. The features and functions are relatively straightforward.
This one hasn't gotten the same level of attention as the others, which is odd. It seems to be getting more traffic. This isn't actually a Kindle Twitter client; it was designed for all mobile devices.