My faucet nut has very little edge to work with. Please help (see picture for clarification). The nut size is slightly smaller than 1 inch. My 1 inch socket couldn't loosen it.
I tend to use Water Pump pliers (10 inch overall). They have cornered faces (120 degrees) which fit four flats of a hexagonal nut, giving a much better grip.
My guess is that the cylindrical part is factory-fitted, and does not even need to be removed. The water flows from the inlet through a slot in the side into the main mixer body. That means it has seals above and below that point, so disturbing it is inadvisable. The skinny grip area is intended to deter you from removing it.
In that case, the underneath side of the cylindrical part will be a brass sleeve two or three inches long (externally threaded), which has the fixing nut and washers for the faucet itself, and a pipe connector fitting, attached, thereby locking the cylindrical part in place (so you will not be able to undo it from the top).
You need to turn off both hot and cold supplies, and work upwards from the pipework: pipe connectors first, then fixing nuts and washers, then lift out the whole faucet.
Anything you need to repair can usually be done from the two ends. The valve part probably pushes down and comes out of the sleeve. If this part does need to be removed (maybe the valve part does come out through the top), it will be easier if the faucet is on a workbench.
You need to get a pair of channel lock pliers.
No home should be without them.
If you can't get the nut off with channel locks, the faucet itself should probably be replaced.
Remember to have the water shut off.