A stray was an animal that was homeless.
In 2369, Kira Nerys compared a computer program that seemed fond of Miles O'Brien to a stray puppy. This analogy later earned it the nickname "Pup". (DS9: "The Forsaken")
In 2370, Jason Vigo described his mother Miranda Vigo's kind nature by saying she was the type to take in stray cats and give her coat to a cold person. (TNG: "Bloodlines")