It shows the rarity. Green glow = rare (rarity number between 8 and 32,+25% stats compared to a common), yellow glow = legendary (rarity number between 0 and 7, +50% stats). The rarity number is randomly rolled between 0 and 255.

Hospital rewards can be rare, but not legendary.

If you equip several items with matching rarity numbers, you get more stats out of them (+10% stats for 2 matching, +20% for 3, +30% for all 4). Mostly relevant for legendaries in the endgame.

Just adding to what Z3n mentioned, if a monster is magichanged (a mechanic you'll learn eventually), they'll turn into a weapon that glows red. It's pretty obvious when you see it (after learning about it), but I've seen people getting confused by it.