Lemon cakes are a kind of sweet baked dessert, made using lemons. They are typically served as small cakes, held and eaten with one hand (as opposed to a larger cake requiring utensils). Lemon cakes are relatively expensive treats enjoyed by upper-class noblewomen of the Seven Kingdoms, where they are a mainstay at the refined social gatherings of noble courts.
In the books
In the A Song of Ice and Fire novels, a running joke is that Sansa Stark is obsessed with lemon cakes. They are a dainty enjoyed by refined "proper" ladies at court, and Sansa can think of nothing better in the world than the paragon of "lady-like" pursuits: a social gathering with major noblewoman wearing extravagant dresses, reclining on feather pillows, sipping tea and gossiping while eating lemon cakes.
Sansa chastises her sister Arya for enjoying "un-ladylike" pursuits, particularly that Arya would rather run off and explore the Ruby Ford with her friend Mycah the butcher's son, when she could be having tea and lemon cakes with Queen Cersei. These lines were cut from the TV show, however, this exchange was used as the audition scene for both Sophie Turner (Sansa) and Maisie Williams (Arya).
Sansa's obsession with lemon cakes is also more understandable in context: citrus fruits such as lemons don't grow in the North, and with medieval methods of food transport and preservation being what they are, lemon cakes are actually a relatively expensive treat for those living in the North. Dorne in the far south is the only region of the Seven Kingdoms where citrus fruits grow, far away from Winterfell.