Vegetarianism is still in its infancy in Cuba, where the idea is basically the more meat there is on a plate, the better. As a vegetarian your staple diet will be rice and beans, eggs, fried plantain, salads, omelettes and pizzas. Cubans often class jamonada (Spam) as not really meat and will often mix pieces into vegetarian dishes, so always remember to specify that you want something without meat (sin carne) and ham (sin jamón).
Vegans will find that they will be extremely limited in what they can eat in Cuba. You’ll generally be better off in paladars, where ordering off-menu is easier and most places serve rice, black beans and root vegetables such as potato and malanga.