Water is dripping, about a drop every other second, out of the end of this thing.
- what is it?
- why is it dripping?
- can I fix it myself?

Water is dripping, about a drop every other second, out of the end of this thing.

The proper name for it is a self piercing saddle valve. The valves are bad for many reasons, including the fact that they often clog and no longer allow water through the line, and they leak.

If you need a valve there to supply something like an ice maker or humidifier, then I'd cut the pipe, add a T, and put a proper quarter turn valve on the line.
To remove it, I'd just cut the pipe right on the piercing and install a copper coupling between the two cut pipes. The steps include: shutoff the water, remove the saddle valve (unscrew the two sides), cut the pipe, clean it, apply flux to the pipe and a copper coupling, fit them together, and solder the joint by heating with a blow torch and working solder into the joint from the opposite side. A skilled plumber should have the entire job done in under 30 minutes including setup and cleanup. For the DIY'er, you'll need the following tools and parts:





Note, no affiliation or specific recommendation with any of the above brands, these are only sample images.
It is saddle valve. Generally they're used to supply ice makers.

It is dripping because the valve is either not closed completely, or because the valve has developed a leak.
Try tightening the valve, but carefully - don't over tighten. If that fails, your next best option is to cap it off. get a 1/4" brass cap from your local hardware/plumbing store:

Also get teflon tape:

Now, shut off the house water supply. Remove the nut holding the tubing onto the valve. Wrap two wraps of tape around the threads on the valve in the direction of threads. Now screw on the cap and tighten until snug. Turn on house water, and open the saddle valve and check for leaks. Then close the saddle valve again.
Cut the valve out and get a sharkbite slip-fitting. Saddle valves eventually leak at the rubber-pipe intersection. Doing it this way doesn't require you to completely drain the line -- a skilled plumber can do a copper sweat job in 30 minutes, but you can do this in 5 minutes with a hacksaw, though a tubing cutter is preferred.
http://www.sharkbite.com/how-to/sharkbite-slip-coupling-copper-or-cpvc-repair/
That is a tap (called a "saddle valve") for something like an ice maker or humidifier. It works by piercing a small hole in the larger copper pipe.
Turning the valve all the way clockwise should close it and stop the drip.
Ideally remove the old, no longer used valve and repair the hole. I do not know how to do that properly, so I yield to a plumbing expert.
Needle valves will always leak eventually. Easiest repair is to get a shark connector SLIP connector. Turn off water, drain pipe. Cut out section with needle puncture with tube cutter, different valves require different sections to be cut out, marked on the slip valve. Push slip fitting in, up and over one end of newly cut pipe, pull back to meet other end Of newly cut pipe, done.
You can do the same with a repair coupling but you’ll have to bend the pipe out of the way to get it in and bend it back and then solder. Not hard, but the “non solder” solution is easier.