Your tattoo artist will have instructions for you. Each artist has his or her own regiman, but usually the bandage should stay on for about two hours.
After you remove the bandage, wash your tattoo with a fragrance-free antibacterial soap, then gently pat it dry. Don't scrub or scrape!
For the first few days, apply an A&D ointment three times a day. Your tattoo artist will recommend a product for you, but check to see that you are not allergic to any of the ingredients in it.
For the next several more days, stop using the A&D ointment and instead apply a frangrance-free lotion three times each day (something like Curel or Lubriderm should work quite well).
After four or five days of this, apply the lotion whenever the tattoo area feels dry or itchy. Whatever you do, don't give in to any urge to scratch it! This can damage your tattoo, and that's not what you want.
For more information on tattoo care, see the other FAQs on 'How To Take Care of Your New Tattoo.'