Compliment comes from French compliment (noun), complimenter (verb), or it may have come from the Italian word complimento, or the Spanish word complimento.  both meaning a courteous greeting or an expression of respect and admiration.  All three were derived from  Latin complementum, ‘completion, fulfillment’. 

The English word compliment, meaning an expression of praise or admiration, was first recorded between 1570 and 1580. However, it was initially spelled and often conflated with its sibling word, complement.  However, according to the Oxford English Dictionary (OED), the distinct spelling of the noun "compliment" first appeared in English around 1655 in the writings of Harmond L'Estrange,  and the verb form first appeared around 1663 in the work of Abraham Crowlley.  

Examples of "compliment" used in a sentence: 
"He has never paid me a compliment. 
"I try to graciously accept both compliments and criticism."
"They complimented me on the way I looked each time they saw me."
"Firstly I compliment you on most of your excellent Spring issue of 'Triangle'. 
"On the one hand, I got a nice compliment from a reader." 
"You can also compliment your loved one on how smart he or she is." 
"They all would make comments to compliment my outfit of the day." 
"Ask yourself what people tend to compliment you on the most."

Don't forget: complement with an e means to enhance, while compliment with an i means in praise of.