Join together two flexible electrical cables
The proper way to do this is to strip back the insulation and twist the two ends together, then solder (weld) and cover in electrical tape.

Without equipment to solder, your next best bet is a butt splice. There are many different kinds of butt splices; this is just an example of a very basic and common one:

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You strip back approximately an inch of insulation on each side, insert them into opposite ends of the splice, then use crimping pliers to crimp the splice on the wire for a tight connection. If you don't have crimping pliers you can use a set of Linesman's pliers, although the crimp won't be as good.  I usually use black electrical tape extending about an inch on either side of the splice to help ensure the connection doesn't get snagged on anything in the future.

The best way really is to solder it, and a soldering iron isn't more than $15 with enough solder to last you for a few years if you end up doing this kind of thing only occasionally.