Glowing wepons?

At level 5 and 20, you'll get a Tag of Sharpness from your Supply Stash. You should have got a Supply Stash from the very first quest-giver at your starting area.
Use the Tag of Sharpness at any mailbox, by accepting quest, to get a glowing quest reward weapon. Quest reward weapons glow blue.
When you get to lvl 60, you might be able to buy or make a Twilight Temple (TT) weapon. Those glow yellow.
The + you see on some people's weapons is because they have refined them at an Elder using Mirage Celestones. You shouldn't worry about refining weapons until 40+, because it could just be a waste before then. I don't think the + on weapons make them glow.
If your weapon has 2 sockets and you have high enough level gems in those sockets, you do get a shimmering spark that orbit your weapon hand.

I think puggy has just about covered it all

if you have 2 or more Average shards of any kind in Armor or Weapon and you get a glow - two of the same kind gives you a specific color (as well as the more important stat boost) while two different shards gives you sparkly effect

see here for more specific info from the wiki

Some more glowy weapons (from the wiki entry above):

Weapons glow for a different reason:
Three-star (purple name) = Blue glow. Obtained from a few quests, boss drops, or rarely (10% chance) from weapon manufacturing.
Legendary (gold name) = Gold glow. Obtained by using materials and a mold (dropped from dungeon bosses) at a Legendary Weapons forge.
Twilight (green name) = Usually a green glow (there are a few blue glowing ones). Obtained by using Twilight Temple materials at a Twilight Weapons forge.
Frostcovered (green name) = Glow green like Twilight Temple weapons. Gotten from Frostcovered City drops.
Frostcovered (gold name) = Red lightning effect on some weapons. Made from "The Past of Frostcovered City" items or Frostcovered Sign