Apple has posted a note warning that Leopard's Time Machine can mess up the Aperture database. The resulting symptom is that Aperture will fail to open. To solve the problem, use the Time Machine system preferences to exclude from Time Machine backups the folder containing the Aperture library (by default, it's inside your Pictures folder). If the problem has already occurred so that Time Machine has already damaged the Aperture library, you will need to rebuild it (hold down Command-Option as Aperture starts up and click Rebuild Now in the resulting dialog).
The question remains: what could be wrong with Time Machine's behavior such that it can have a bad effect on Aperture's library? Reader Tom suggests that Time Machine "may do an update on a changed file before or after the Aperture metadata has been updated"; but this seems a little unsatisfactory as an explanation, especially since Time Machine is (or should be) merely copying files, so how can it have an effect on those files? As usual, Apple is not vouchsafing any details.