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.
Olenna and Margaery Tyrell share some lemon cakes with Sansa Stark when they meet her in the gardens of the Red Keep to learn more about King Joffrey Baratheon, now Margaery's future husband. Sansa reveals that Joffrey is "a monster".
In the Eyrie, Sansa eats lemon cakes with Lysa. Sansa then queries how they make the cakes, as it would not be possible to grow lemons in the Vale of Arryn. Her aunt informs her that Littlefinger had brought three crates of lemons from the capital, knowing how much Sansa is fond of lemon cakes.
In the books
In the A Song of Ice and Fire novels, a running joke is that Sansa is obsessed with lemon cakes. They are a delicacy enjoyed by refined "proper" ladies at court, and Sansa can't think of anything better in the world than the paragon of "lady-like" pursuits: a social gathering with major noblewomen wearing extravagant dresses, reclining on feather pillows, sipping tea and gossiping while eating lemon cakes.
Thus in many ways lemon cakes come to symbolize Sansa's concepts of what normal, glamorous court-life is like: she wanted nothing more than to have lemon cakes and tea with Cersei while chatting about her proposed marriage to her son Joffrey - but this eventually turns into a living nightmare, as Joffrey kills her father and has the "honorable" knights of his Kingsguard beat her in front of the entire court, with Cersei unable and unwilling to restrain her son. When Olenna and Margaery meet with Sansa, they give her lemon cakes, treat her with kind words, and give her the basic respect befitting the daughter of one of the major Houses of the realm - symbolizing that they maintain the refinement and decency that the nobility of the Seven Kingdoms are supposed to possess, but the Lannisters lack.
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 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.
Robert Arryn also loves lemon cakes, perhaps because "Alayne" does. She promises he can have as many as he likes, and he demands a hundred; "Alayne" thinks she would like to give him a hundred spankings instead.
For the feast held for the competitors at the tourney of the Gates of the Moon, Lord Nestor’s cooks prepare a splendid subtlety, a lemon cake in the shape of the Giant's Lance, twelve feet tall and adorned with an Eyrie made of sugar. The cake required every lemon in the Vale to make, but Littlefinger has promised that he would send to Dorne for more.
Jeyne Poole, Sansa's best friend, shares her love for lemon cakes. In "Theon I" sample chapter of the unpublished sixth novel, she still impersonates Arya, at Theon's advice. Mors Umber questions her, and she nearly gives herself away by mentioning the lemon cakes Gage used to make for Sansa (there was no reason for the real Arya to say that, since she - in constrast to Sansa - was never particularly fond of that food).