How do I repair or replace this fridge water line connector?
I've seen that piece called a Water Connector Union Fitting and a Tubing Coupler.

Verify the size of the tubing, but if it's the normal 5/16" to 1/4" it should be easy to get a replacement at any larger hardware store, such as this.  The ones I've seen (that look just like yours) don't require any tools, other than possibly a blade to cut a bit of the tubing off at the end if it's calcified - you just insert the tube.  

You might also check to see if your tube just got pulled out a bit - it's possible it doesn't require replacement.