Bullets and caps are/were separate.
Percussion caps were used on handguns, rifles and shotguns during the early to mid 1800's.
I attached another photo of some newer produced caps.
See the attached photo of the revolver, the area within the red circles is where the percussion cap would be placed, those are called nipples.
A powder charger would be poured into each chamber from the face (front) of the cylinder. Then a bullet was placed in the chamber and pressed into place, usually with a rod that is attached to the gun. The cap placed on the nipple and you are ready to shoot.
Cock the hammer and pull the trigger. The hammer would hit the percussion cap causing a small explosion within it. That would travel through a small hole into the chamber and ignite the powder charge. The result would be your projectile/bullet being fired. That is if all went right. Check out the little gif I found, its on a rifle but the idea is the same.
Without seeing your bullets I can only guess as to what you found. The "rusty" one could be lead with something adhered to it or it could be a newer bullet that is actually steel jacketed. If you can find a way of posting pics we can better determine what you found.
