He shall remove the ash of that which the fire consumed… (Leviticus 6:3)
Terumas hadeshen was the daily process in which the kohein removed the ashes of the sacrifices from the altar. This was done every morning at dawn, earlier on holidays (which were busy times in the Temple). The job was assigned to a kohein by lottery except on Yom Kippur, when it was performed by the Kohein Gadol. After all the ashes were gathered, they were taken outside the camp.
The purpose for this mitzvah was to increase the prestige of the Temple by removing the ashes. Aside from “taking out the trash,” removing the ashes helped the altar fire to burn better.
The kohanim did not perform this task wearing their usual working clothes. The Talmud (Yoma 23b) uses the analogy that a butler shouldn’t pour wine for the guests while wearing the same stained apron he had on while making chili in the kitchen. (Clearly, I have paraphrased their analogy.)
This mitzvah applies to male kohanim when the Temple service is in effect. In the Talmud, it is discussed in tractate Yoma starting on page 20a. It is codified in the Mishneh Torah in the second chapter of Hilchos Tamidin. It is #30 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.