A "trattoria" is a folk or popular restaurant with a homely atmosphere, usually run by a family. In it you'll find a typical Italian cuisine.

there is no single word which can convey this word, let's try a few. Keep in mind that "trattoria" is sometimes used to give the impression of "home cooking", family-run, family atmosphere, when in fact the place is a and remains, for all practical purposes, a restaurant. Choosing the right name depends on the image the restaurant wants to convey to the public:
"trattoria" (I would consider leaving like it is; do a web search and you'll see)
family restaurant
regional cuisine
home cooking
diner (except that it is so heavily connotated in US English)
take out (in the French sense of "traiteur"; again search the web)
La trattoria restaurant (I found one mexican site by this name);
Regards
(what's the contest?)
Paola L M