What can I do with an enum variable?

Really what you're doing there is declaring a variable inline with the rest of the definition of the enumeration. It's equivalent to:

enum paint_colors { RED, GREEN, BLUE };
enum paint_colors colors;

Often, you'll see a typedef associated with the definition:

typedef enum _paint_colors { RED, GREEN, BLUE } paint_colors;

Which lets you use the enumeration more like the built in types:

paint_colors color;

So the answer to your question is that colors is a variable of type enum paint_colors, and can be used any way you think is appropriate for such a variable:

colors = RED;
if (colors != GREEN)
{
    colors = BLUE;
}

And so on.