Elizaveta Héderváry represents the Republic of Hungary, and is described as the manliest character in the series in addition to a "reliable older sister type". She was once married to Austria (thus forming Austria-Hungary) and lived in his house alongside Chibitalia.

Gilbert Weillschmidt represents the Kingdom of Prussia. He has a younger brother, Germany, who he affectionately (or not) calls "West". He loves fighting, and avoids marriages; he is also quite ego-centric.

Their relationship is represented in most fanart as one from their childhood. Elizaveta believed herself to be a boy until hitting puberty, and it is often portrayed that they went on many adventures together as children.

Although mainly a fanon pairing due to their interaction when they were kids opposed to concrete evidence in the present, Elizaveta did divorce Roderich; who knows what the future holds? :)

TRIVIA:

+ They are referred to as "Gilbo/Eli" on Pixiv (by Japanse fangirls)
+ Gilbert is 6" taller in the series; however, Elizaveta is usually the taller one in their childhood portrayals.
+ Another popular pairing is Gilbert and the genderswap of Elizaveta; coined "boy Gary/Pru" on Pixiv.
+ Elizaveta is often seen with a frying pan (mostly used to hit Gilbert with).
+ Glibert is shown with a yellow chick, mostly on his head or shoulder.

Source: Hetalia Wikia & fangirl knowledge.