A croquette is a small patty, ball or cylinder of puréed salmon, potatoes or some other item that is usually breaded and deep-fried.
Croquettes are bound together with eggs and a starch such as potatoes, rice or bread crumbs, and sometimes a basic white sauce.
The most basic kind is potato, which is basically mashed potatoes with some egg yolks mixed in to hold it together, then rolled in breadcrumbs and fried until crispy.
These are great, because almost anything deep-fried until crispy is great. These days, crab cakes and salmon cakes are typical examples of croquettes.
But in the old days, chefs used to make croquettes out of anything they could get their hands on, and it was a great way to use up leftovers, which was always one of the biggest priorities for chefs in the old days, because the old days I'm referring to are the days before refrigeration.
Thus, ingredients like meat, game, veal, poultry, foie gras, offal (such as brains and organ meats), seafood (fish, lobster, mussels, oysters), vegetables, cheese and pasta were made into croquettes. In addition to savory croquettes, sweet ones, made with fruit and/or nuts, combined with pastry cream, were also common.
The standard procedure involved chopping up the main ingredient and then cooking it along with diced mushrooms and truffles plus a bit of wine, such as madeira. Next, the mixture would be combined with a béchamel sauce, or sometimes a basic brown sauce, plus a few egg yolks, and then cooked a bit more, before turning the mixture out onto a dish and layering with butter.
After it cooled, the filling would be divided into portions, dipped in egg, rolled in breadcrumbs and deep-fried until golden brown.
I don't know about you, but a croquette made with wine, béchamel, truffles, mushrooms and chopped up meat or offal or oysters or brie cheese sounds like something to seriously write home about. And the point is, you can totally make that. Whether it's with leftovers, or something you go out and buy especially for this purpose, there's really nothing to it.
Simply choose your ingredients, chop them up into a kind of hash, combine with egg yolks and béchamel, roll into cylinders, bread and then fry.
See the picture above? It's called arancini, which is a type of croquette made in Italy using leftover risotto (but you can use ordinary rice) plus mozzarella cheese, fresh parsley and roasted red peppers, then breaded and fried. Check out this article on the standard breading procedure, and this one on deep-frying.