(First off, it's "das Inserat".)
"das Inserat": an ad in a newspaper, saying e.g. that I am selling my furniture, looking to buy old coins, looking for a new partner. I'd imagine it to be more likely from a private individual, and it would appear in a certain section of the paper, like "Buying/Selling", in a column together with other ads, not as a colourful stand-alone thing.
"Die Annonce" would, in my book, be from a private person, too, but I don't think the word is very common.
"die Anzeige": can be the same as an Inserat, but just as well advertisement by a company in a newspaper (of the colourful stand-alone kind) or on the Duolingo website. Especially in a paper, you can specify what kind of Anzeige it is, e.g. Heiratsanzeige (wedding announcement), Stellenanzeige (job advertisement), Verkaufsanzeige (when you sell something)...
"advertisement" could cover other things/translations as well (but not being a native speaker, I'm not sure about the details, e.g. if you'd prefer "advert" over "advertisement" when talking about an ad in a newspaper), e.g. "(die) Werbung": anything that advertises a company/product, e.g. a commercial break on TV, a TV commercial spot, a leaflet, a billboard, (again) an ad in the paper, a guy dressed in a mascot suit on the street... "Das ist doch Werbung für dich!" = "Why do you expect me to pay your band for playing your music at my wedding? After all, it's advertisement for you!" ...or "die Ausschreibung" = when a company says that they are looking to hire someone for a certain job