Can I access TimeMachine data from a machine other than the one that was being backed up?

You can navigate your backup drive in the Finder. Mount it, go to the Backup folder > [name of machine] > [name of volume] > latest > Users > [user name] and restore the files you want.

(You can also choose a date instead of latest to restore earlier versions of the files/files that have been deleted.)

You can use the Time Machine UI to restore from the backups of a different Mac by option-clicking the Time Machine icon in your menubar or right-clicking the Time Machine icon in your Dock, and selecting "Browse Other Time Machine Disks". (See question 17 from the Time Machine FAQ for more details.) You'll still need to navigate to "Latest" (or an earlier backup) as zevlag describes in his answer if you choose to do this.

If you do elect to use the comfort of Time Machine to choose what to restore, you'll want to read question 16 from the FAQ: "How can I restore a file/folder to an alternate location?"