How do I claim tax relief on donations? (US Only)

In summary:

Charitable contributions or donations can help US taxpayers to lower their taxable income via a tax deduction.
To claim a tax-deductible donation, you must itemize on your taxes.
Your donation must meet certain guidelines to qualify - for example donating to a qualified organization.
The amount of charitable donations you can deduct may range from 20% to 60% of your AGI

This article summarizes claiming tax relief on donations. (Please note this article is for US donors. For UK tax benefits on donations, please see What is Gift Aid?)