To repair this wire on the mower can I simply solder it together or will I need to replace the part? What is it called?

 

It looks like a high temperature cut-out.
Did the black cables break off of it?
Does the mower run when the two black wires are joined together?

 

Looks like thermal cutout rated at 130 °C As MrChips said.
Good luck.........

 

It would be more difficult to solder than to just replace it with a similar device, ideally one with leads already attached that you can connect with your existing wires. Maybe those existing wires have connectors on the far ends?

Getting the device too hot during soldering will blow it, and to me it looks like the one-shot kind. That's why the original construction uses a crimp. Some can cycle repeatedly but I don't think this is one of those. These things only cost a buck but are hard to find. If you can find it on Amazon, for example, you'll usually have to buy 5 or 10 of them.