Here's what you'll need:

• Soldering iron
• Electrician's solder
• Flux (no-clean flux is good, rosin flux is good, acid flux is bad)
• Wire cutters
• Wire strippers
• Heat shrink tubing (the kind with heat-activated adhesive is nice)
• Cigarette lighter or heat gun
• Wet rag to clean soldering iron tip

Here's what to do:

• First you need to get your wires ready. Make sure you have enough wire to make your connection and have at least 3 inches of slack after the wires are routed the way you want. Once you've got this, strip about 1 inch off the end of each wire. Resist the temptation to twist the stripped portion of wire so that it's neat. You should have something like this:
  
  
• Now get a bit of heat shrink tubing, at least 4 inches long. I like to use heat shrink tubing with a heat activated glue in it -- this when heated, it shrinks and glues itself to the wire, which creates a tight, water resistant seal, which is what you want. Anyway, get your heat shrink tubing (it should fit easily over the wire, with enough room to fit over a solder joint!) and slip it over one of the wires. Get it as far from the joint as possible, so that it doesn't shrink while you're soldering.
• Don't forget the last step! You'll feel pretty silly if you go solder the wires and then realize that your heat shrink tubing is not on the wire.
• Now is a good time to plug in your soldering iron, if you haven't already.
• Now stick the stripped ends of the two wires together, as close as they go:
  
  
• Twist the stripped part a bit: grasp one end of the connection with your thumb and forefinger, so that you have a good grip on one wire and about 1/4 inch of the stripped part end of the other wire. Grab the same bit of the other wire with your other hand, and twist the two. Get one or two full twists.
• Bathe the stripped part of wire with flux. I recommend rosin flux or a 'no clean' type flux. Flux pens are handy. Get lots of flux on it; it should be dripping. Don't do this on your coffee table; do it on the workbench or the garage floor, or something.
• Make sure your soldering iron is tinned -- there should be a bit of clean solder on the tip of the iron. If there is not any solder there, or there is but it is not clean, you should clean off the tip and put some more solder on it. To clean the tip, get a wet rag, bunch it up so you don't burn yourself, and wipe the tip of the iron.
• Solder! Have the end of your piece of solder touching the wire, and heat the wire next to it rather than the solder itself. The flux will steam and bubble and might splatter a bit -- be careful! If your wires are well fluxed, the solder will get sucked into the wire. Keep applying solder until the entire 1 inch connection is totally coated. Don't use too much solder, or you will have trouble getting the heat shrink over it. If the solder bubbles on the surface of the wire instead of getting sucked in, you either don't have enough flux, or the wire is dirty. In either case, dump on more flux. This isn't the most efficient method of cleaning wires but it seems to work. You can also try heating up the wires a bit more with the iron, but be careful not to melt the insulation or whatever surface you're working on.
• When you're done, and you have a nice neat connection, unplug your soldering iron and let the wires sit until the connection cools. This is important because if the connection is still hot when you try to slide the heat shrink tubing over it, the tubing will shrink and make life difficult.
• When the connection is cooled, slide the heat shrink tubing over it so that there is at least 1.5 inches of tubing on each side of the connection:
  
  
• Now get a cigarrette lighter or heat gun and heat the tubing. Always heat the tubing from the center and work out, never start from one end and work to the other. If you are using a lighter, don't let the flame touch the tubing or it will melt.