You should be able to fit a 660Ti in that case, not 100% sure. Regarding the 6-pin, that's a power connector for the video card. You have to connect the 6-pin from the PSU(power supply unit) to the 6-pin header on the GPU. Before you buy the card, I am pretty sure you are still using the stock PSU and that will not be enough to power the GPU let alone the whole computer.
Most computer companies sell their computer with low end PSU's that are meant to only run the parts that they originally built it with. Most likely your going to have to buy a new PSU and install that before you install your GPU.