Cultured Marble
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Use newspaper and painter's tape to protect any surfaces you don't want splattered. Don't forget the faucet and handles.
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Wet sand the countertop with 1,000-grit sandpaper. Don't press too hard or you will break through the gel coat that seals the surface of the vanity top. Rinse off the countertop and dry it thoroughly.
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Use Meguiar's Medium-Cut Cleaner (or a similar product) and your power buffer or hand buffer to smooth out any remaining scratches and give the counter a fresh shine.
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Apply a generous amount of marble polish to the countertop, and rub it in with a clean shop cloth. Let it dry until it is opaque, and then buff it with clean paper towels for an extra layer of protection.
Laminate and Vinyl
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Use an abrasive cleanser like Ajax to "sand" fine scratches out of Dupont Corian, and then go back to cleaning it with a gentle cleanser.
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Use tinted vinyl sealer (found at flooring shops) to repair large scratches in vinyl. If you cannot find vinyl sealer, use silicon caulk, either in a color that matches your countertop or clear. The latter won't hide the scratch, but it will keep water from soaking through and damaging the wood underneath the vinyl.
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Wipe down the countertop thoroughly with a damp cloth or sponge.
Granite
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Fill deep scratches (any that are deep enough for a fingernail to fit into) with epoxy tinted to match the granite. You can get this at home improvement stores. Let it dry according to the manufacturer's guidelines.
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Use your wet grinder to buff the epoxy and the area just surrounding it. Use very light pressure so you don't damage the granite.
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Clean away any dust with a soft, damp shop cloth.