Take the record to a professional for cleaning. It is also possible to professionally clean the record yourself by soaking it in a very weak dish soap solution, running a soft toothbrush around the grooves, rinsing it with distilled water and wiping it with a microfiber cloth.
Scrape the surface of the record with a recording stylus or a wire brush. Follow the path of the record, being sure not to create new scratches.
Heat the record in a microwave for a minute and a half (one minute for 45rpm records). An alternative to this option is to heat the scratched section with a lighter for a few seconds; this may melt the scratch enough to even it out.
Tape a quarter or other weighted object on the top of the record player's needle. This will push the needle against the record with more force. Over time, this may eliminate the scratches in some cases.
Convert the record to a digital file by recording it into a computer, as a last resort. Use an audio editor to digitally remove the scratches from the audio.