faucetinstallationplumbingremoval
I purchased this faucet and it seems like an easy enough installation. The bottom of this new faucet looks like this: 
Seems like all I have to do is screw something in and it will attach the faucet to my counter top.
My old faucet: 
Doesn't seem like it has the same mechanics. This is what I see under my kitchen counter top: 
How do I get that off? I've tried twisting it as if the shaft is threaded but I don't think it was loosening at all.
See the threaded tube? There's a nut threaded on that, which holds the plate in place.
Use a Basin Wrench (or any wrench you can fit in there) to loosen the nut (rotate the nut anticlockwise to loosen it), and the plate will slide out.
If the nut is stubborn, and doesn't want to budge. A good shot of penetrating oil can loosen things up.
I drew a small black circle around the area of that tab where you can slip a utility knife point underneath of it and pop it out it should come out pretty easily.
Best Answer
The black collar seen on the bottom picture is what is keeping the faucet on. It is a plastic nut that screws on the threads you see to clamp the faucet onto the countertop.
That is what you need to unscrew with the basin wrench (or any other that can reach).