From Dark Brown to Red Hair
A: Of course it is possible. However, you need to be sure that you are realistic with your hair goals so as to prevent damaging your hair overmuch in the process of achieving the color you desire. The basic steps involved will likely be first lightening the hair to the desired level and then applying the desired color.
Before you decide to make this step, I do want to recommend that you become well informed about your options and the processes involved in coloring the hair. We have several articles on hair coloring that can help you understand what's involved in the coloring process.
You also need to be sure that you would look good with a red color. If your natural color is a warm tone (with a red or orange base color) you will probably be fine, but make sure you are certain of the choice before you try it.
When making dramatic color changes, especially combining lifting the natural color with adding a new color tone, I always recommend seeking a professional's services. Your salon professional will have access to color swatches of hair that you can hold against your skin to visually confirm that the color you want will suit your skin tone.
The professional can also best analyze your hair's current condition to determine if your hair will withstand the processes involved and can suggest alternatives and advise you on the safest way to get the results you want (or at least, results with which you can be happy).
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See also:
The different hair color levels
How to choose the right shade of red hair color
How to lighten dark red hair
How to avoid fast red hair color fading