A disc is composed of a tough outer layer of cartilage (annulus fibrosis) that surrounds a softer gel-like material (nucleus pulposus) in the center. Discs also contain proteins and water. A disc bulge is when the disc comes under pressure or degeneration and bulges out but all the contents of the disc remain within all these fibres. Usually the precursor to a disc bulging is some degeneration happening causing the disc to weaken. LIke dessication (the disc drying out with age causing it to weaken), or excess pressures on the disc (like lots of extra body weight), a spine out of alignment, or some accident. When the bulge happens it can press on other surrounding structures. Think of a disc bulge like a water balloon, when you squeeze it and how the water is still inside but you can distort the balloon. They say up to 80% of disc bulges are asymptomatic and cause no pain.
A herniated disk results when a crack in the tough outer layer of cartilage allows some of the softer material inside to protrude outside of the disk. Herniated disks are also called ruptured disks or slipped disks, however the accurate name for them is herniated. This disc material excapes and presses and irritates the other structures in the spine it comes in contact with. You can get pain not only from pressure, but also something called chemical radiculitis which is just an irritation from the material that shouldn't be outside the disc. There are degrees of herniation, from the simple tear with a little leak to a complete fragmentation of the disc. The hernated disc is more damaged than a bulged disc.