Which blood vessel begins at the lateral border of the first rib, and terminates as it passes by the tendon ot the teres major muscle?

The closest answer to this question would be the axillary artery. The axillary artery is the extension of the subclavian artery, which is an extension of the brachiocephalic artery (if you're starting point is at the right side of the subject's body) or the aortic arch (if you're starting point is at the left side of the subject's body.)

One can also argue that the answer can be the subclavian artery because if you look at anatomical atlases, it is the subclavian artery that goes over the lateral border of the first rib. But because the question includes the teres major muscle, it is the axillary artery that actually terminates close to its level. Please refer to pictures online to help you further.