The presidential security service is tasked with protecting the President of the Twelve Colonies. They were hand picked by Captain Lee Adama from civilians in the Fleet. Responsible and answerable only to the President, they have no allegiance to the Colonial Fleet. They consist of former police, security personnel, or people with some kind of firearms training.
The PSS on Colonial One is a distinct entity from the military personnel aboard Galactica and Pegasus. Lee Adama himself had intentionally designed their organization in this way as he thought they should have a separate command structure from the military organizations, so the lines of authority establishing who was under whose control would not be confused.
When Commander William Adama sent Marines led by Colonel Saul Tigh and Lee Adama to Colonial One to arrest President Roslin, they engage in a tense standoff with the security service. Eventually, Apollo pulls a gun on Tigh, then Roslin orders Adama and the security service to drop their weapons, preferring that she would leave peacefully. (Kobol's Last Gleaming, Part II)
Note
The term "presidential security service" is a descriptive term, not a canonical designation. Ron D. Moore discusses them in his podcast for "Kobol's Last Gleaming, Part II."