A service branch was a subdivision of Starfleet's organizational layout.
Nancy Hedford blamed the medical branch of the Starfleet for not providing her with the proper inoculations prior to her arrival on Epsilon Canaris III in 2268. (TOS: "Metamorphosis")
When Julian Bashir questioned Luther Sloan in 2374 whether he was really from Internal Affairs, Sloan explained that while that it was a competent but limited department, he was not, stating rather, "Let's just say I belong to another branch of Starfleet Intelligence. Our official designation is Section 31." (DS9: "Inquisition")