1. Press v
2. Highlight the dwarf you want to cut wood
3. Press (p)ref
4. Press (l)abor
5. Use +/- to go to the 'woodworking' category
6. Press enter.
Is the 'woodcutting' skill highlighted for this dwarf? If it is not, then woodcutting is not enabled on him and he will not cut wood. Enable/disable using (enter). If woodcutting is enabled on him, go to the z->stocks menu and check you have axes. If you're not sure because the accuracy is too low, I recommend appointing a bookkeeper and putting him on max accuracy (n)obles -> scroll to bookeeper -> (enter) -> set precision. This will confirm for sure whether you have an axe available for use. Alternatively, if you are sure you have axes available (not forbidden), also check the dwarf's (i)nventory at step 3 for said axe.
Also check if the dwarf you want to cut wood has a proper path to the wood, and you have designated the wood for cutting using the (d)esignate -> cut down (t)ree. If all of the above are satisfied, and he's still not cutting wood, check if he's on break or doing something else using the (j)obs menu (look for his name). If he's sitting around with 'No Job' and all of the above are satisfied, you've waited for a minute or two, and he's still not doing anything, then you have a more difficult problem to work out, perhaps a bug (unlikely though, the game's gotten pretty good on this sort of thing).
The above process works for pretty much every type of 'dwarf not doing job' situation where you suspect it's not because of the dwarf being a lazy bastard.
The above