No matter what project you're working on, you'll eventually have to connect some wires together.
There are a few different ways to connect small wires. Some require only a bit of duct tape and you'll need special tools for others. Let's look at six different methods, one of which allows you to easily detach the wires in the future.
Warning: do not connect mains voltage (120/240V) wires using any of these methods!
Before you can connect wires together, you have to strip the shielding first. If you are only doing this once or twice a year, any kind of sharp edge will do the job. I've actually used a utility knife to strip hundreds of cables, but you do need to be careful and it takes up a lot of time.
Looking back, I really regret not buying a wire stripper sooner. I kept thinking to myself that this is the last time that I'll have to strip the cables by hand, so there's no need to waste money. But then something else came up and then another thing and another thing and it would just never end. Do yourself a favour and just buy one right away. They're not that expensive and even cheaper ones work better than cutting your fingers.
Next, having something to hold the wires in place will make your job a lot easier as well. Anything that can hold a wire will do, even two empty paper rolls. Just cut a groove and you're good to go.
Of course, you can also 3D print one of many holders like this one.
Naturally, a proper soldering stand is your best bet if you'll be doing this often. You don't have to spend a lot of money on it. The one I have was under 10€ and even comes with a magnifying glass and integrated light.
Finally, if you have to connect wires in place, then I'd recommend printing this handheld holder. This will be especially useful for joining wires that are already part of something and you can't easily remove them.
Tape it together
The first method is quick and easy, but I wouldn't use it for anything but a very short term solution until you can make a better connection. Begin by twisting the wire strands together as tightly as you can.
Finish off by applying electrical tape around the twisted part for insulation. The tape will also provide a bit of strength by holding the two wires together.
A much better way is to solder the wires together. Twist the strands as before, use something to hold them in place and use a soldering iron. The easiest way is to use a flat tip which you then place under the wires. This will ensure that the wires get hot enough and make a good connection with the solder. Don't push the solder against the iron to melt it but only touch the wires instead.
You can then use electrical tape as insulation, or you could buy special heatshrink tubes. Just place it around the splice and use hot air gun over the area. The tube will shrink and insulate the wires, but also hold them together.
Solder and seal connectors
The next method is a combination of the previous one. You can use tubings like this that are called solder and seal connectors.
Simply place it around the splice and heat it up. The solder in the middle should melt and flow over the wires, and the tube should shrink and hold everything in place.
It's an easy and effective solution, but the connectors are a bit more expensive. This method was actually used during the WW2 to quickly repair broken wires on the field.
Push-in wire connectors
Using push-in wire connectors, you simply push in both wires and you're done.
Of course, if you're using a stranded wire, you'll have to crimp a connector to one end so that you have something solid to push in.
This is probably the best option if you have to connect multiple wires at once, as you can buy connectors with more than two slots.
You're probably already familiar with Dupont connectors as they are used on a lot of electronic equipment, including extensively on Arduino and PC's. They're great for quickly connecting things together. They're also perfect for replacing different parts, as you can swap anything in seconds.
The downside is that they are more difficult to assemble and require special tools and connectors. You will need to buy both male and female connectors, as well as the plastic cases and a special crimping tool.
The housings come with different number of slots, which makes it easy to connect multiple wires at once.
Lastly, you also need a special crimping tool that can handle Dupont connectors.
Sadly, ordinary crimpers (on the right side in the photo below) have different dies that usually don't work for Dupont connectors, so you will need to buy a dedicated Dupont crimper.
The fan on my printer is a great example of when you should use Dupont connector. If it ever fails, I can easily replace it in the future.
Quick splice connectors
The last method is fast and easy and you don't even have to use any tools at all. You simply insert the wires on both ends, without cutting the insulation, and push down the pins. It will cut through the shielding into the wire and create an electrical connection.
Of course, nothing is perfect. These connectors aren't exactly cheap and they will "damage" the wire after making the connection. If you want to reuse them on the same wire again, you'll have to cut away the end first. If the wire was already short to begin with, you might not have enough of it left.