The word "complement" comes from the Latin word complementum, meaning "that which fills up or completes".  It derives from complere ("to fill up"), itself from the intensive prefix com- and the verb plere ("to fill"), ultimately stemming from the PIE root pele- meaning "to fill".  Complement (with an "e") is a noun for something that completes or enhances another, or a verb meaning to complete or enhance. 

According to the Oxford English Dictionary (OED), the noun complement was first used in English in the Middle English period (1150-1500) with the earliest evidence from 1398 in a translation by John Trevisa.  The entry notes that the earliest recorded use for the verb complement was in 1617. 

Examples of "complement" in a sentence: 
"The wine complements the meal very nicely." 
"Nicky and Kai complement each other well when they work as a team."
"Abdul's technical knowledge complemented Sarah's writing skills." 
"Following the recent recruitment drive, we now have a full complement of staff." 
"The buzz she enjoyed inside the classroom was complemented by her love of sport, especially in combat."
"Overnight, grassroots efforts that once complemented federal work have taken on outsized urgency."
"We complemented each other along the way," she said.

 Don't confuse complement (with an "e") with compliment (with an "i")!  
Complement (with an "e") is a noun for something that completes or enhances another, or a verb meaning to complete or enhance. 
Compliment (with an "i") is a noun for praise or admiration, or a verb for expressing praise. 

Remember: Complement with an "e" means "enhance", and compliment with an "i" means "in praise".