A stray was an animal that was homeless.
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")
In, 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")