Damage reduction is written as "Damage reduction X/quality". If you hit someone with DR with a regular attack, reduce that damage by X points unless the attack has the right quality. In D&D 3.0, the quality was a weapon with a specific enhancement bonus most of the time, and better weapons trumped worse ones. In D&D 3.5, the quality can be a specific material (usually silver, cold iron or adamantine), an alignment, a damage type (slash/pierce/bludgeon), or magic/epic, and there's no hierarchy where some things are better than others. Most spells bypass DR entirely, though there are elemental resistances that work similarly against those.
Fast healing is simple. Every round, unless you're dead, you regain X hit points.
Regeneration is a bit more complicated. First, regeneration turns most of the damage you get into subdual/non-lethal damage. Second, each round you remove X points of subdual/non-lethal damage. Some damage types will not be turned into subdual, depending on the creature in question. Trolls, for example, take normal damage from fire and acid.
Regeneration is generally considered stronger than Fast Healing, because regeneration basically makes you unkillable. On the other hand, fast healing doesn't give you an achilles heel the way regeneration does.