What they do
Anchor rollers provide a rolling surface that makes it easier to drop and weigh anchor without banging up your topsides. Anchor roller/mounts allow you to stow your anchor conveniently after you weigh it.
How they work
Anchor rollers make it easier for you to weigh anchor by hand since you can place your body where you can pull most efficiently. They also create a fair lead for the anchor rode so it doesn't chafe against your stem fitting or chocks. The rolling action reduces chafe as the nylon line stretches.
Anchor mounts cradle the anchor on the bow of the boat so that it's always ready for deployment (a basic tenet of good seamanship). They are practically required when your boat is equipped with a windlass, since they guide the anchor rode to the windlass gypsy, and keep chain from gouging your hull and deck.
Powerboats commonly place anchor mounts on their centerline, since they have no head-stays. Sailboats, with stem fittings, head stays, furling drums, bow pulpits, etc. should have mounts placed 4"-8" to the right (or left) of the head-stay. Wooden blocks may be required to raise the mount above the toe rail. It should be parallel to the centerline of the boat, however, and aligned with the windlass gypsy.