How can I repair my damaged kitchen counter?
That's a laminate countertop. You may want to try to use a repair compound like SeamFill or FormFill Laminate Repair - the damaged spot is small enough where it may work. You'll need to remove the blistered material with a sharp knife before applying the compound. Instructions for application, complete with '70s style cartoons, can be found here.

If the results aren't satisfactory, you will need to cut out a section to install an inset cutting board, or re-laminate the countertop - neither option is simple or easy.