With the wires you want to join twisted together and held in place, pick up your soldering iron in one hand and your solder in the other.
Touch the tip of the soldering iron to the wires and keep it there.
The wires will begin to heat up. At some point over the next 2-10 seconds (depending on how hot your iron is) the wires will be hot enough to melt the solder. You can touch the solder to the wires (not to the tip of the iron!) periodically to see if it's hot enough. It's tempting to just touch the solder to the tip of the iron and melt it right away, but don't! You will end up making what's called a cold solder joint. This occurs when you melt the solder around the joint, but you aren't melting the solder into your joint or onto your components to make a good connection. It's much better to wait the few seconds and melt the solder onto the hot wire itself.
If you touch the solder to the wire and it begins to smoke and melt, the wires are hot enough. Add the tip of your solder to the joint as necessary. You want to introduce enough solder to cover the wires, but not so much that you create a big glob of solder at the bottom of the joint.
Once you've got what you think is enough solder on the joint, pull the solder away and then remove the soldering iron. If you're using a gun style soldering iron like I was, release the trigger to turn it off. If you're using the kind that doesn't have a trigger the iron will stay hot, so just place it back into the holder.
Here is a video of wire being soldered. The whole process happens pretty fast, and isn't nearly as complicated as you may have expected.
Here is a close-up video of the same process.