How and whether you can fix your broken molding will depend entirely on how badly it is broken. If your molding is just chipped, fixing it should be a cinch. If you somehow tore an entire piece out of your molding, though, you may as well just replace it.
If your molding is only slightly broken, say you chipped it or tore a small piece out, you can repair it. Invest in some wood putty and fill the gaps you made. Allow the wood putty to dry overnight. The following day, repaint your molding. The molding should look as good as new.
If your molding has serious damage, tear it down and put up new molding. This will be easier than trying to fix it. You don't need to tear down the molding around the entire room; you can just replace the molding on the wall with the damage. Pop the broken molding off the wall by sliding a chisel behind it and gently lifting. If the wall beneath the molding needs repainting, this is the time to do it. Replace the old broken molding with new, plaster covered foam molding. This stuff is cheap and easy to use, and once you have it painted you won't be able to tell it apart from your old molding as long as you can find the same style.
To install the new molding, cut a piece to the size you need and affix it to your wall with joint compound. Plaster coated foam is so lightweight that you probably don't even need a friend to help you. If you are putting up an especially long piece of molding, secure the center of the piece with a few nails so that it doesn't droop. Clean up the excess joint compound, let it dry overnight and paint the molding the next day.