まぐろ (also written as マグロ and 鮪) is the Japanese word for thunnus, a specific kind of tuna. It refers to both the living fish and the food. Traditionally, まぐろ also referred to billfish because billfish was considered to be a close kind to thunnus. Because of this, even today まぐろ can also refer to billfish.
ツナ comes from the English word tuna and it refers to drained and flaked tuna (not necessarily thunnus), which is mainly used for tuna salad. I do not know if drained tuna before being flaked is also called ツナ or not.
シーチキン is the product name of canned ツナ made by Hagoromo Foods Corporation. (The Japanese Wikipedia states that it is named after chicken because its taste is similar to chicken.) But this product is so common in Japan that ツナ in general is sometimes called シーチキン.
Edit: Revision 1 contained a mistake about the meaning of the word まぐろ.