I suspect most of us just slice wire in the middle, and connect what we need, using solder to make a permanent connection, and putting heat shrink material around the joint and shrink it like zaxarias's post.
But if you are still a solder newbie, you can get cold splice wire taps that allow you to connect cables without soldering (you do need to use enough force to do the splice with large pliers). Adafruit sells these: https://www.adafruit.com/products/1494, but you probably can find similar things elsewhere. Note, you want to buy splices for the size of wire you are using. The Adafruit version can do 22-26 gauge wire, which typically matches what most people use.
Alternatively, if it is power or ground wires, you can use standard breadboards, and they have red/black columns that allow you to plug in separate wires to the board. Even if they aren't power/ground wires, you can plug up to 5 wires into a row on the breadboard, and they are are connected. You do have to make sure sure if you are driving signals through the wires, that the total current is less than or equal to the amount the microprocessor can send through a given signal pin. Here is a tutorial on breadboards: https://learn.sparkfun.com/tutorials/how-to-use-a-breadboard
If you want to start making your own wire that are custom sized, you can get crimp connectors that you can attach to stranded wire. I've used the connectors from Pololu.com: http://www.pololu.com/category/39/cables-and-wire. I find crimping the female end on a little challenging with just needle nose pliers, and one of these days I need to get a proper crimping tool.