"Burning" is the process of writing data to a writable or rewritable optical media disc, like a CD-R or DVD-R. The term comes from a disc drive's laser burning a series of dots into a layer on the underside of the disc. These pits, and the gaps between them, contain the digital data encoded on the disc.
A professionally-pressed CD or DVD has a series of pits stamped into a layer of reflective metal under a protective layer of plastic. When an optical disc drive reads the disc, a laser focused on the underside reads the flat areas (which reflect the laser light) and the pits (which scatter the laser light), then decodes that data. A writable disc has an extra layer of transparent dye between the plastic and the metal. The recordable disc drive uses its laser to burn a series of dots into the dye layer. When read later, these dots scatter the laser light the same way the pits stamped into a pressed disc do.
Rewritable discs are burned just like writable discs are, but instead of a layer of dye, the drive's laser burns the data onto a reflective layer of a phase change metal alloy. This layer starts in a crystalline form that light can pass through to the reflective metal layer underneath. The disc drive's laser burns data to this layer by heating the alloy past its melting point, which turns opaque as it cools. The disc drive can erase a rewritable disc by using its laser to heat the alloy back to its crystallization point.
NOTE: Burning data to a disc is more delicate than writing data to a hard disk or flash media. First, a computer organizes data into the correct file system for the disc. It must also keep that data in a buffer to write to the disc at a constant pace. If this buffer runs out of data, called a "buffer underrun," the burn will fail. A rewritable disc can be erased for another attempt, but a write-once CD-R, DVD-R, or DVD+R disc is ruined and destined for the trash.