Is the GT 620 good for gaming?

Only for very old or simple titles at low settings. It cannot run modern games at playable frame rates, and integrated graphics on a recent CPU will usually beat it.

Does it need extra power from the power supply?

No. The card runs entirely from the PCIe slot, so no separate power connector is required.

Will it fit a slim case?

Yes. This is a low-profile card, so it fits slim and small-form-factor desktops.

How much memory does it have?

This version has 2 GB of DDR3 memory. The DDR3 type is much slower than the GDDR5 used on better cards.

Does it support two monitors?

It has display outputs for more than one screen, so it can drive a dual-monitor desktop setup.

Is it better than my CPU's integrated graphics?

Usually not. On any recent Intel or AMD processor, the built-in graphics are faster, so the card mainly makes sense for older systems without a working output.