Over time, a countertop can be damaged or discolored. In most cases, you can simply sand back the damaged area (though only for solid wood countertops, not veneered ones).
Burning and staining can look unsightly (Image 1), but these problems are easy to repair in two steps.
Sand the area back to the bare wood, making sure not to create a depression (Image 2). Then use a cloth to remove all traces of dust and debris.