(Disclaimer: Language is subjective. YMMV.)
nostalgia
From Greek roots meaning roughly a longing to return home, nostalgia used to mean homesickness, but now means a longing for the past.
gay
It used to just mean "happy". Now, of course, it denotes a minority sexual orientation.
democracy
From Greek meaning "rule by the people", a system where political power is directly in the hands of the people. Nowadays it's also synonymous with the republic, where power is held by elected representatives. (I don't like this change. America is not a democracy.)
atheist
From Greek meaning "godless", a person without belief in a god. Now almost exclusively (IME) used for people who have rejected the idea of the existence of a god, as opposed to people who have merely never been introduced to the concept.
down
Apparently this all-purpose direction word used to be a longer word meaning something like "from the hill"; i.e., downhill. Now I know why "downs" such as Watership Down are actually "up".
Christian
Originally an insulting term used for those who "followed the Way" (of Jesus Christ). Now used with quite a bit of overweening pride by people who never learned not to take the Lord's name in vain.
typewriter
It used to also mean the person who ran the typewriter as well as the machine itself. Then we got the word typist. (Then we got rid of all the typewriters.) Apparently the same thing happens with other appliances, like the dishwasher.
basilisk
It was an animal first described as a snake with a kind of crown marking on its head, hence the name 'basilisk' -- 'little king' in Greek. The creature has become much more elaborate over the ages, with all kinds of quirks, like having a cock's head, and being hatched by a toad.
Semantic drift rocks.
(But it makes making an a posteriori conlang really tough.)
ShadowNode: atheist is entirely Greek, without a drop of Latin blood in it. (Well, -ist might be Latin, but the Latins borrowed it from Greek anyway.)