I have been using an FTP client almost daily since I started working as a web developer in 2001. For me, it's absolutely essential to my job function and something that I couldn't imagine living without.
At Myriad Interactive, we use CuteFTP Professional for file transfers and remote text editing. Its interface is very intuitive and the lockstep folder navigation (which keeps local and remote folders syncronized) is a must for developers. The built-in text editor is also outstanding. HTML files are color-coded and you can perform find-and-replace operations across multiple open files. The only downside to CuteFTP Professional is the $59.99 (per user) price tag.
While my endorsement obviously goes to CuteFTP Professional, not everybody needs a fully featured FTP client, they just need a way to upload large files or to make a backup of their website. If you fall into this group of users, the following programs should do the trick...and they're free.
CoffeeCup Free FTP