Heartwarming things move and inspire you. If you want to get a job writing greeting cards, you better learn how to write heartwarming messages.
When you're sick, little gifts and offers of support from your friends can be heartwarming, and stories that end with lost dogs reunited with their families, or good people living happily ever, after are also heartwarming. The origin of this adjective is fairly obvious: it describes things that "warm," or give a feeling of affection and pleasure to, your heart. A related word is warm-hearted, which dates from the 16th century.