While this tutorial has provided a fair amount of information about wire and cables, actually making the choices of which to use boils down to some simple steps:
1. If you are prototyping low power circuits, you can use breadboards, breadboard wire and/or some smaller gauge wire between components.
2. For low-power projects, batteries will most often have a JST connector. Low voltage power supplies most often have 2.1mm barrel connectors.
3. For high current projects such as using many LEDs (NeoPixels, DotStars, etc.), add up the maximum current of the LEDs and the project, then size the power supply to meet that current draw (and possibly a bit extra for headroom).
4. For flexibility, use stranded wire. For ease of soldering small wires to circuit boards, use solid-core wire.
5. For wall (mains) power, always follow the electrical code in your region and country. ALWAYS exercise caution as mains power can cause great harm if a bare conductor is touched. NEVER plug bare wire into a wall socket.
6. Make good electrical connections. Soldering permanent projects forms a good electrical connection and a fair mechanical bond. Specialized end connectors can provide a good mating surface with other components.
7. Insulation protects your wiring. After running your wire and making connections, be sure wiring is not exposed where it could touch other wires or small fingers.
8. Many cables come premade from companies like Adafruit. If you need a specialized cable, you can obtain the wire and connectors for a custom application.
9. If you use or reuse magnet or transformer wire, it has a clear laquer coating which you need to scrape off the ends when connecting to other circuit parts or a battery.