"a throb, a beat, a stroke," especially a measured, regular, or rhythmical beat, early 14c., from Old French pous, pulse (late 12c., Modern French pouls) and directly from Latin pulsus (in pulsus venarum "beating from the blood in the veins"), past participle of pellere "to push, drive" (from PIE root *pel- (5) "to thrust, strike, drive").
Extended usages, of feeling, life, opinion, etc., are attested from early 16c. The figurative use for "life, vitality, essential energy" is from 1530s.
"peas, beans, lentils; the esculent seeds of any leguminous plant," late 13c., puls, from Old French pouls, pous, pols and directly from Latin puls "thick gruel, porridge, mush," which is suspected of being (perhaps via Etruscan), from Greek poltos "porridge" made from flour, or both the Greek and Latin words might be from the same source (compare pollen), which might be a loanword from a non-PIE Mediterranean language or an as-yet-unknown PIE root.
"to beat, throb," as the arteries or the heart, early 15c., pulsen, from pulse (n.1) or else from Latin pulsare "to beat, throb." Related: Pulsed; pulsing.