Ok so here is the relevant picture with some highlighting.

The "removable slot" (backplate) needs to be removed and the single screw holding it in place will then hold the card in place when installed.
The card will just slot into the "PCI-Express" slot and the green clip with click into place by itself when the card is fully seated.
When handling computer components its always best to make sure you do not have any static electricity which could discharge into the computer during contact. You can touch something grounded like a radiator to discharge yourself or use rubber gloves.
Do not work on a computer while it is still plugged in or the power is on.
Cards within your budget and on this website will be either a 8400GS or 5450. Might be worth starting a debate about which is better for minecraft to decide which to go for, I assume memory would not be an issue with that game so the 512mb versions should narrow it down a little more.
Length should not be an issue as they are pretty minimal cards by design anyway.
Anandtech's website seems to having some issues right now but they have a good comparison chart for card performance. I dunno if they list these cards but if they do I think you can find the comparison tool here - http://www.anandtech.com/bench/GPU/88

The "removable slot" (backplate) needs to be removed and the single screw holding it in place will then hold the card in place when installed.
The card will just slot into the "PCI-Express" slot and the green clip with click into place by itself when the card is fully seated.
When handling computer components its always best to make sure you do not have any static electricity which could discharge into the computer during contact. You can touch something grounded like a radiator to discharge yourself or use rubber gloves.
Do not work on a computer while it is still plugged in or the power is on.
Cards within your budget and on this website will be either a 8400GS or 5450. Might be worth starting a debate about which is better for minecraft to decide which to go for, I assume memory would not be an issue with that game so the 512mb versions should narrow it down a little more.
Length should not be an issue as they are pretty minimal cards by design anyway.
Anandtech's website seems to having some issues right now but they have a good comparison chart for card performance. I dunno if they list these cards but if they do I think you can find the comparison tool here - http://www.anandtech.com/bench/GPU/88