Like so many other things there is no one 'best' framework.
1) Joomla is a content management system (like wordpress), not a framework.
2) cakePHP is probably the most popular php framework and uses prototype/scriptaculous for some ajax support
3) Some other PHP frameworks I know of are:
Zend - Developed by the guys who handle PHP
CodeIgnitor - Don't know a lot about this.. but it seems to be picking up quite a following
Symfony - Probably lesser known than CodeIgnitor.. but also picking up steam