An orifice was an opening in one's body.
In 2369, Data observed that Devidians had an orifice where a humanoid's foreheads would be, which ingested particles, possibly food. (TNG: "Time's Arrow")
In 2370, a replicant of Miles O'Brien, frustrated with Doctor Julian Bashir's thorough physical, noted that Bashir had " poked into every orifice of my body and created a few new ones." (DS9: "Whispers")
In 2373, The Doctor noted that a macrovirus had created an orifice from a Garan miner's glandular tissue through which to escape. (VOY: "Macrocosm")