How do I tell what sort of solder I have?
If it's not labeled you can't determine its makeup easily. The answer depends on why you want to know though. It probably doesn't matter if you're doing this for hobbyist things and not commercial.

For example, lead-based solders will melt at lower temperatures than lead-free versions and are recommended for hobbyists because it's easier to work with. In a professional environment most products have to be made with lead-free solder though. 

If you're just soldering for a hobby side project don't worry about it. If you have trouble getting the solder to flow than purchase some leaded solder and it should be easier.

Edit:

If you're just strictly trying to determine whether or not it is lead-based or lead-free than you can use its shininess as a good guess. Lead-free tends to be shinier than leaded. If you want to know the exact makeup percentages like Sn vs. Cu vs. Ag then the chemistry exchange might be a better place.