to recline at a table to dine; more basically, to lie stretched out
In the Gospels this word almost always means to recline at a table for a meal. In the Greek and Roman world, guests at a formal dinner did not sit on chairs; they lay on their sides on low couches set around a table, propped on the left elbow with the right hand free to eat. To be "reclining" was therefore simply to be at dinner, and "the ones reclining" was an ordinary way to say "the dinner guests."
That everyday posture carried a great deal of social meaning, which is why the Gospel writers reach for it so often. Reclining was the posture of a free and honored person, and where you reclined marked your rank. So when Jesus reclines to eat with tax collectors and outsiders, or when a disciple leans back against his chest at the last meal, the seating is doing real work: it pictures acceptance, honor, and closeness at the table.
The word belongs to a small family of Greek verbs for the same act, and they shade into each other: κατάκειμαι is the older word for the same reclining posture; ἀναπίπτω stresses the moment of settling back onto the couch rather than the settled state of already lying there; συνανάκειμαι points to the company you share the couch with; and ἀνακλίνω and κατακλίνω are the words for seating someone, making a guest recline.
Once, though, the word simply means to lie down. In Mark's account of a young girl who had died, it describes her lying on her bed, not a guest at a feast.