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If you purchase at any of the local home stores, you are going to find basically stranded copper wire, and solid copper wire.
I can't imagine you would want solid wire, so I assume you plan to use stranded wire.
With stranded copper wire, the only differences in what you'll find locally is size.
There is tinned copper available, which is copper wire that has a coating of another metal over it. But I doubt you would find that for sale at a home store. You wouldn't likely find it as easy to work with anyway, as it's not as flexible.
When soldering stranded copper wire, you need to make sure you get the proper solder, or it won't stick to the wire.
The color of the insulation has nothing to do with the elemental makeup of the wire. It's strictly for identification purposes only, relative to the function of the wire. For example, black would be a hot color, as would red. White is a neutral color, and green is a ground color.
There are small rolls of project wire available, or, you can purchase by the foot from the wire rack.
Standard stranded copper wire is known as THHN. That's the insulation rating of the wire.
For something as small as your LED light strips, I wouldn't plan on using wire any larger than #14. If they have smaller wire, you could actually go a size or two smaller. However, when we advise our customers about wire sizing, we generally go by the amperage rating of the fuse or breaker that feeds the circuit the wire will be used on. If you are using the LED strips on a 15 amp circuit, that would mean you need to use #14 wire, as that's the amperage rating for that size of wire.