Elocution is (1) a style or manner of speaking, and (2) the art of public speaking. A locution is a word or a phrase or the act of saying a word or phrase. Locution is so often used in place of elocution that many dictionaries now list the words as synonyms in some uses, but the words generally remain separate in well-edited writing.
A third, rarer word, allocution, refers to a formal or authoritative speech or address.
Examples
When she heard me developing a broad Yorkshire accent, I was sent off to elocution lessons. [Daily Mail]
[P]erhaps the oddest locution Gingrich used was when he called the plan “right-wing social engineering,” on par with “left-wing social engineering.” [CNN International]
In dubbed versions, no one is going to fuss about Vin Diesel or co-star Dwayne “the Rock” Johnson’s elocution. [Globe and Mail]
Romney repeated the “we’re going to hang him” locution once more . [The Atlantic]
Sam Bryant has the lead role of Eliza Doolittle, a Cockney flower girl who is given elocution lessons so she can present herself “as a real lady”. [Marlborough Express]
Blame monogamy for that awkward locution–there’s no plural for “Mrs.” [Wall Street Journal]