Variac is the trademark name of a variable autotransformer. A variable autotransformer is a single-coil transformer in which two portions of the same coil are used as the primary and the secondary. Variable autotransformers are used for the same purposes as regular transformers, but handle lower voltages.
After performing a repair on a piece of electronic equipment, it is best to slowly power up the device to avoid burning up the replaced parts. You will often discover multiple part failures one at a time. Using the Auto transformer to slowly power up the repaired unit can prevent damaging the replacement parts.