It's called a tap connector (obvious now I've found it), and you should be able to unscrew it. You can see one here
It's the bottom nut you need to be turning. It's part of a joint that attaches the pipe to the tap.
Once you have this off you should be able to remove the tap as normal.
If it's stuck then something like WD40 or penetrating oil might help. Failing that carefully heating the nut with a heat gun might expand the various parts enough to loosen them. Take precautions to avoid getting direct heat onto the bath (it appears to be acrylic).
A more drastic approach would be to take a hacksaw to the thread to separate it that way.
Where does the hose in that image currently go? Is that an existing dishwasher inlet or does it go somewhere else like a laundry closet or to a sprayer/sink?
Also, we need to know what type of fitting the dishwasher inlet will be. Most that I know of are threaded flex hose.
The part you show does not look like what you want, if you just want to screw it into your existing fitting. This part designed to be inserted into the middle of an ordinary copper pipe using compression fittings that join to an unthreaded copper or nylon pipe end, providing a ball-valve connection in the middle of that pipe suitable for a washing machine. You CAN use this, especially since you have a shutoff valve (which appears to have been installed in much the same way), but you must permanently alter your plumbing by cutting some part of the pipe in half to put this fitting in.
What you probably want is a male-male-female NPT equal tee or Y fitting, allowing you to screw in a second hose exactly like the one you have now. Something like this (from the same site) may fit the bill:
If you can find it in nickel-steel, it'll last a little longer, but this plastic fitting should be perfectly fine. Simple, near tool-less (just a pair of slip-joint pliers or a small strap wrench to tighten down the hoses), and undoable.
Best Answer
In the UK this would be called a compression waste fitting. There is a rubber ring that goes around the pipe, and is squashed by tightening the nut, in order to achieve a watertight seal.
The pipe should go past the nut, into the fitting, until it stops. This would probably be an inch (25mm) or slightly more.
To fix the joint:
One alternative point: If the water is collecting at this joint, it is not necessarily this joint which is leaking. I once took a joint apart and put it back together three times, trying to make it watertight, before I realised that the joint above was the one leaking, and the water was just collecting at the lower joint.
However, given that you say the pipe comes out of this joint easily, it probably is this joint which is leaking.