What kind of terminal is this, and how can I fix it?
It is called a spade connector, or spade lug terminal. You are holding a female connector. 

You can purchase these at an electronics supply shop. As you might expect, they are available in a bewildering variety of styles and sizes. You might want to bring one of your connectors for identification. 

Some are designed to be soldered onto the end of a wire, and others are made to crimp on using a bespoke tool. Either way, I think you'll find that heat-shrink tubing makes an excellent protective insulation.