How do I determine the absolute configuration experimentally?
If your compound has been synthesized before, you can measure its optical activity and compare it with the data in the literature.

If your compound hasn't been synthesized before, it can be possible to differentiate between two enantiomers with NMR by measuring the spectra of your molecule in a chiral environment. Here's a review article on how to resolve enantiomers with NMR.

Otherwise, diastereomers can be differentiated easily. By looking at your molecule, you can (sort of) predict which spectrum it should give rise to by using the concepts of enantiotopic, homotopic, heterotopic and diastereotopic protons. We had to analyse and solve 2D-NMR spectra (1H and 13C) of diastereomers last year, and to a beginner, I have to say it was a challenge.

As always, it boils down to forming diastereomers.

EDIT

Take a look at these diasteromers:

 

(I've got finals coming up), it's easy to see that the hydroxyl will shift the ethyl's CH2 peaks towards lower field (higher ppm) in the second isomer. The interpretation of chiral media NMR spectra would have to be similar. My impression is that the acquisition of chiral media spectra is more involved, but the interpretation isn't. I might be totally off track, though.

Someone might want to share a more thorough answer.