1620s, "a sentinel," agent noun from round (n.) on notion of "one who makes the rounds." Sense of "chronic drunkard or criminal" is from 1854, on notion of one who is habitually in and out of jails or bars. Rounders, a baseball-like game, attested from 1828.
round
round robin
roundabout
roundel
roundelay
rounder
Roundhead
roundhouse
roundness
round-table
roundup