First full suspension bike?
It's the bushings that have worn out and need replacing, these are the steel sleeve inside the shock eyelet.
These are pushed out (with a special tool or carefully with a socket), and a new ones pushed in to replace them.
They are a sacrifical part, so that your shock eyelets doesn't wear.
Take a look here (you only need to replace the bushing):
http://www.tftunedshox.com/info/mount_kits_bushings.aspx
You don't have to buy from them, any bearing supplier will be able to supply you a generic bushing's (if you can supply the sizes).