When examining a weapon, there are two things being displayed about the weapon that aren't mentioned in the manual. First, there are a couple of black arrows, top and bottom, layered over the condition bar of the weapon. What do they represent? They are also present on armour.

Second, there is an bullet icon to the right of the image of you holding the weapon. There is a number next to the icon as well. What does this represent?

The first thing you're talking about is the weapon's condition, if your weapon's condition is to the right of the arrows, your weapon does not need to be repaired. If the condition is to the left of the arrows, your weapon is not performing as well as it could be, and it is recommended that weapon be repaired.

I think the second thing I think you're talking about is how much ammo you have.

The arrows on the condition bars tells you when the item starts to perform poorly due to lower condition. A weapon deals less damage once it gets below 75%, an armor protects less when it goes below 50%

The icon on the weapon shows what skill is used to use the weapon - bullets for gun, a lightning-thingy for energy weapons and a grenade for explosives.
The number tells you the minimum skill needed for not having penalties when you use the weapon.

Thanks. I guess I don't have any weapons that require skill yet. All my bullet numbers still say zero.

Yeah, the first guns you come across doesn't have any requirements. It starts with 9mm smgs, 10mm pistols and cowboy repeaters with a requirement at 25.