The first set of damage numbers is conventional damage, while the second (in blue) is magical. There are melee weapons and bows that can do some melee/ranged damage and some magical damage; the regular damage gets a bonus from strength (melee) or dexterity (ranged), while the magical portion gets a bonus from intelligence. The name of most weapons will indicate what damage they do; 'normal' weapons do regular damage, 'balanced' do both, and 'magical' do predominantly magic damage.
After finding an enchanter (possible from about character level 10 or 12), you can also add enchantments to weapons for normal and magical damage, if they have one or more enchantment slots (up to 3 are possible, though rare).
