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I'm forgetting my D&D rules.

When I see a magic weapon with damage listed as, for example 2D4 +3, what does the "+3" mean? Does that mean I can do 3 points additional damage per level of my character? Or, is that a flat 3 extra points of damage per hit?

When it says something like +3, it should mean that 3 extra damage is added to the damage you would otherwise deal.

Flat damage per hit. A 2d4 +3 will do 5-11 damage per hit, plus strength bonus.

Weapons-based damage never depends on the level of the character, except indirectly by how often they hit (Thaco). However, many damage spells are level-dependant, and are written as 1d4 +1/level or 1d6/level up to 10d6, for example.