They're a line of machines filled with plastic capsules called Gacha Balls. When a buyer puts 200円 ($2.67) inside and turns a handle, one capsule will fall out to be taken. The items to be found randomly packed inside are made exclusively by the company Takara Tomy.
These are quite similar to capsule machines in other countries, but differ by usually being bigger, and always having much higher quality items.
While they sound nearly the same at first, Gachapon may not be interchanged with the term Gashapon. The latter, while the same style and method, is a term copyrighted by the company Bandai, and applies exclusively to items made by them, instead. It should be noted, at the time of writing this, that Bandai has produced absolutely no items for the Hetalia series, and so no Hetalia gashapon items exist.