Now that you've bought all your supplies, taken an airbrush class, practiced thousands upon thousands of circles and squares (when you take an airbrush class, you'll understand what I'm talking about...), and are an airbrush MASTER, you can learn how to cover ANY tattoo. Like it was never even there. Until you take a shower, that is.
Actually, I like to cover tattoos with alcohol-activated paint, because it is basically waterproof and sweat-proof. During class one day someone colored in my tattoo with alcohol-activated paints, and because I didn't bother to remove it, it lasted for nearly 4 full days.
Step 1: Color-Correct
Before starting any airbrushing, make sure you are wearing an apron and your client is wearing a cape and has the skin or clothing around the tattoo covered with paper towels to avoid spilling airbrush paint on them. Then you're going to begin by color-correcting every different colored section of the tattoo.
I have a super hardcore tattoo of California on my inner arm that I decided to cover. Since it is only one color (and just an outline and super tiny), it was very easy to color-correct. In order to color-correct you'll cover an area with the color that is directly opposite it on the color wheel. For example, cover purple areas with yellow, pink areas with green, etc. Black ink reads as blue, so you will use orange (on darker skintones) or yellow (on pale folk like me) to correct any black areas.
Try to keep the airbrush paint restricted to just the area that the tattoo covers. I did a crappy job and went way outside the lines because I was airbrushing myself with my left arm and it was quite a challenge.
Step 2: Begin Adding Skin Tones
After you've airbrushed a thin layer of color-correcting paint on your tattoo, begin adding skin tones. I like to mix a bit of yellow in with a skin tone that is slightly deeper than my own first, and then follow that layer with successively lighter and lighter colors until it matches my skintone perfectly.
Be sure that you're airbrushing in thin layers and allowing each layer to dry completely before starting another.
Step 3: Finished!
If you've done this correctly, your tattoo has now VANISHED! Something that I forgot to do that would've been really helpful in blending the coverup would be to take a chip brush, dip it in a bit of brownish-colored alcohol-activated paint, and draw your thumb back across the bristles while holding it near your coverup. This creates freckles and irregularities in the skin to make it look more natural.