The Department of Thoughts appears to be a building inside Patrick's brain (which appears to be purple). It has a blue sign on top that states the name and blue doors and stairs.
Role in the episode
It is shown that there are lots of mini Patricks in the Department of Thoughts when Patrick cannot read the Chum Bucket slogan, "Chum is Metabolic Fuel." By the time Patrick had his resignation, the Department of Thought was completely burned up and exploded, yet somehow Patrick was able to make a slogan when he was hired by Mr. Krabs.