Why does some of a person's tan wash away?

I assume you are referring to spray tanning.  If this is correct then chances are the solution applied to your skin had a bronzer in it to give you an instant tan.  This bronzer stays on your skin until the first shower.  You then wash and the bronzer goes away and what is underneath is the DHA developed tan.  It takes roughly 6-10 hours for the DHA to develop the tan under the bronzer which is why most technicians tell their customers not to shower in this time frame.  Using a DHA friendly soap like Bodyguard: DHA Friendly Body Wash can help maintain your tan.  Remember that spray tans are topical and not to exfoliate after you receive your tan.