Pierce fitting leaking
Does anybody know what this is called? 


This is commonly called a saddle valve; they are frequently (and notoriously unreliably) used to tap into cold water for supply to refrigerator ice maker, hence the 1/4" compression type outlet commonly used for 1/4" diameter tubing. IMO you should not try to plug it, you should get rid of it.

If you do not need to supply an icemaker, cut below it and solder fittings to restore the faucet supply valve. If you do need to supply an ice maker, put a tee in the line and get a high quality 1/4 turn valve with the same compression type outlet.