The short answer to this question is yes, you can give blood after getting a tattoo. There are a few caveats which are important to know about, however, and you may want to be aware that many blood banks reserve the right to refuse donations even if donors are technically eligible to give, based on the interview performed before blood donation takes place. Although it can be disappointing to be told that you cannot give blood, this is done to protect the safety of the blood supply, and you can always return later to offer blood, because blood and plasma are always needed.
In some regions of the world, tattoo artists and their facilities are closely regulated, and subject to mandatory inspections by health department personnel. In these areas, you can give blood after getting a tattoo right away, without a waiting period, although you should still disclose the fact that you have a new tattoo to blood bank staff. If you aren't sure about whether or not tattoo facilities in your area are subject to such regulation, contact your health department for more information, or call your blood bank and ask specifically how long they want you to wait before giving blood.
In areas where tattooing is not as closely monitored, people are asked to wait for 12 months to give blood after getting a tattoo. This waiting period ensures that the donor is free of any potential blood-borne diseases which could have been passed on unsanitary equipment. The vast majority of tattoo artists and studios take your safety (and theirs) very seriously, and even without regulation, they typically autoclave all tools, use fresh sterile needles for tattooing, and observe other safety precautions. Since they are largely self-regulated, however, blood banks like to be on the safe side.
If you are very committed to giving blood on a regular basis and you also enjoy getting body art, you may want to make a habit of giving blood shortly before you receive new tattoos, and scheduling sessions for big projects close together, so that you can start the clock on your waiting period as early as possible. You may also find that some blood banks are more friendly to tattooed people than others; staff who are not familiar with the infection control procedures used in tattoo shops may reject you out of fear, while staff who have been informed about the safety of modern tattooing may be more than happy to take your blood.
When you give blood after getting a tattoo, you may want to wait at least a week, even if you are allowed to give blood right away, as tattoos often cause low-level inflammation. Giving your body a chance to recover will ensure that your blood passes the screening procedures used to test donations. Blood banks also ask that donors not use them as screening facilities for STDs and blood-borne diseases; many public health facilities offer such screening tests for free, without compromising the safety of the blood supply.
The same holds true for new piercings, incidentally. If you are pierced in a thoroughly inspected shop, you can give blood right away. If the conditions are at all questionable, however, you will likely be asked to wait 12 months.