A digital perm is a perm that uses hot rods with the temperature regulated by a machine with a digital display, hence the name. The process is otherwise similar to that of a traditional perm. The name "digital perm" is copyrighted by a Japanese company, Paimore Co. Hairstylists usually call it a "hot perm."
A normal perm basically requires only the perm solution. A digital perm requires a (different) solution plus heat. This type of perm is popular in several countries, including South Korea and Japan.
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Difference between a normal perm and a digital perm:

The biggest difference between other perms and a digital perm is the shape and the texture of the wave created by the digital process. A normal perm, or "cold perm," makes the wave most prominent when the hair is wet, and loose when it is dry. The hair tends to look moist and as locks. A digital perm makes the wave most prominent when the hair is dry, and loose when it is wet. Therefore you can create the dry and curly look of the curl iron or the hot curler.

Digital perms thermally recondition the hair, though the chemicals and processing are similar to a straight perm. The hair often feels softer, smoother, and shinier after a digital perm.