The most common sugar used in cooking is granulated white sugar, which, depending on crystal sizes, ranges from superfine to regular to coarse.
Powdered sugar is just granulated white sugar ground to a powder (often with an anti-caking agent added, but not if you make your own in a blender).1
Brown sugar (light or dark) is white sugar where some of the molasses is added back in. The darker the sugar, the more molasses added and the stronger the flavor. If a recipe doesn't specify whether you should use light brown sugar or dark brown sugar, just use whatever you have on hand or prefer.
Turbinado (also called raw sugar) and evaporated cane juice are both sugar that has been processed less and thus has retained some of the molasses normally removed to make white sugar. Despite some clever marketing, these "natural" and less processed sugars aren't really healthier and should still be consumed in moderation.2
Store sugar in airtight containers. If your brown sugar still gets too hard to use, you can put a slice of bread or some apple slices in with it and check on it after a day or two (the airtight container remains key though!). If you need to soften it immediately, put it in a glass bowl and stick it in the microwave along with a glass bowl filled with warm water. Don't heat it for more than 20-30 seconds or so at a time or you run the risk of melting it. Break up large chunks with a spoon as you go.3