If something is heartfelt, it's sincere and warm. It's clear from your elderly neighbor's heartfelt invitation that she genuinely wants you to come over for tea and cookies.
When you're profoundly grateful to your favorite teacher, you'll offer her your heartfelt thanks. And when your best friend's dog dies, you'll give him your heartfelt sympathy. Things that are heartfelt are deeply emotional, believed or felt very strongly. The word dates from the 18th century, from heart, with its Old English root heorte, "heart, spirit, or mind."