Downloading a previously purchased app to iTunes on another computer
It is possible, there are a few steps required.

The other computer must be authorised to access that Apple ID.

On the iTunes store page on the right hand side, there is a Purchased option, click on this and all previous apps, movies, music, etc purchased will be shown. You can then download it from there.

Apple has a help page about this that can be viewed at http://support.apple.com/kb/HT2519. It is the bottom section.