Ok, so I recently got a Lenovo Thinkpad T440s, and the graphics are terrible. The actual laptop runs perfectly- it’s quite quick for its price, but the gaming performance is unplayable. If I purchase an external graphics card, will I be able to play games?
That system is not for gaming, it's a business class system, wrong thing to get if you were looking for gaming on a laptop.
For external video cards you would need to connect it to an internal mini PCIe slot or M.2 slot if that system has one, then run the ribbon cable from the case, with the video card sitting on the desk connected to an external power supply. Past all that, the external video card stuff is not very reliable to work with all laptops, works for some not for others. There is no list that I remember seeing as to what laptops work with what external adapters.