The tiles that run vertically down the sides of some fireplaces are also known as cheek tiles. You will need to remove the fireplace surround and the fire grate in order to change a tile.

Removing the fireplace surround is a fair undertaking and you might prefer to repair minor cracks instead. To repair minor cracks in fireplace tiles do as follows:
Clean the broken fireplace tile well and dry.
Wipe PVA or epoxy glue into the crack to repair discreetly.

You might like to paint the crack to fit in with the rest of the tile using enamel paint.

Removing a fireplace surround

First look for and locate the fixings for the fireplace surround.

Removing a marble fireplace surround

Marble fireplace surrounds are often fixed to the wall with hidden ties. The fixings will usually be covered with plaster. Once you have located the fixings dig into the plaster to free them up. As you undo the fixings please be aware that most marble fireplace surrounds consist of several pieces, so be prepared to support the different pieces, as they come apart.

Removing a wooden and cast iron fireplace surround

Wooden and cast iron surrounds are usually fixed on with screws. Cast iron fireplace surrounds will often have lugs at the sides.

Removing fireplace tiles

Fireplace tiles are often fixed in a metal frame held on with nuts. Once undone remove tiles.

If you are not replacing, but planning to keep and repair, the next step will be to clean the tiles.

Cleaning fireplace tiles

Lead stains can be removed using paint stripper or thinners.
Household bleach will remove stains where the tiles have cracked.

Repairing fireplace tiles

The next step will be to glue the tile together. You can do this using PVA glue or epoxy.

If pieces have chipped off the tile you can make up filler by mixing talcum powder with epoxy resin.
Once tried this can be painted using enamel paint.

Re-positioning the tile.