First before I ask my question.. I am not completely sure if what I have is called a memory card reader.. it's a drive under my cd drive that has sd slots.. usb slots.. and some other slots.. whatever..

How do I remove this so I can place it inside another computer?
All I see is a black cable connected to it.
And if I disconnect it, I still don't know what to do since I don't see any screws.. and I'm not sure if I'm supposed to just push the drive out.

I'm not sure you can do what you want. Another computer probably does not have the wiring to accept it.

USB memory card readers are dirt cheap (typically $15)... I suggest just picking one up for use on your other computer.

Listen to RickJ, he speaks truth. Go get a USB memory card reader. They are cheap. The other computer's motherboard probably does not have the connections needed to hook it up.