My doctor told me I had a bulging disc in my back. Will I need surgery?

Not for a disc. Bulge as theses are fairly normal findings on an imaging study and are not sources of the pain. So there is no indication to operate on a disc that is only bulging as its only finding or presentation.
Maybe. You are a surgical candidate if 3 criteria are met: 1.) pain that lasts more than 6 months and that has failed to respond to nonsurgical measures. 2.) there is a surgically "correctable" problem. In the case of a bulging disc, one does not know the answer to this until a discogram has been performed. Yes, bulging discs can be sources of severe pain. 3.) you determine that you can't live with it.
No. Odds are no! Most of the time, surgery should be a last option. Most people have bulging discs. It depends if the bulge is "touching" import structures as to what your treatment may be. The bulge in itself may not be causing any problem.