What are some good CHEAP graphics cards that I can install on my laptop?

I have a Lenovo Z570 Ideapad. I use this laptop for gaming, and it only has the problem of a really old graphics card. I found that my laptop may or may not be able to switch graphics cards. I also do not know which graphics card I should get because it needs to be the same size as the one I have now.

For a laptop its better to get a cpu with integrated graphics which is a apu which is a cpu and a gpu combined to save space since a graphics card does not fit in a laptop

You can't change the graphics card on a lap-top. It is integrated.

Some laptops do have special versions of graphics cards in them, but they are either integrated into the motherboard or are specially made for that laptop.
It's highly unlikely you could change a laptop graphics solution.