We must ask ourselves: How does our body detect pain? The nerves detect pain and relay the message to the brain. The other thing is that it sometimes depends on the nerve or nerves, you see, some detect pain and some don't. In fact, most of our nerves don't actually detect pain at all.
What is the purpose of our disc? The discs allow us to twist, jump, run, and more, and have some shock absorption while doing those things. Not only that, they act as spacers between spinal bones. This allows for an opening so the nerve roots can exit off of the spinal cord. In actuality it is the brain, to spinal cord, to nerve roots, to smaller, and to smaller nerves, to every part of the body.
What happens when a disc gets injured? We can have a few types of injury. A disc can degenerate, it can bulge, it can tear, or it can rupture. A degenerative disc, or arthritic disc, is one that has flattened and possible bulged out. Analogous to a flat tire, the disc is similar, it flattens and bulges out. Now it does depend exactly where that bulge is; this may tell us if we get pain.
The nerves exit off of the cord and out of the spine in the back side of the column. We refer to that as poterolateral. A bulge, rupture, or tear here may be the most painful. This is in comparison to a disc that bulges more toward the anterior may cause much more pain.
Many people have degeneration or arthritis and are walking around with no pain. There is some other symptomatology besides pain of someone with degeneration. That may include morning stiffness or decreased range of motion.
There are some sensory nerve fibers in the discs themselves also. Hence a disc tear may give pain. When a disc burst or tears because of a herniation you may get pain for both reason. The inner disc is referred to as the nucleus, the outer part is the annulus.
With a bulge that inner disc may push out on the annulus causing a "bulge". In a rupture or herniation, the nucleus breaks through the annulus.