A yearning is a strong desire. If you have a yearning to travel in Africa, it means that you dream about it and intensely hope that one day you'll end up there.
A strong longing for something is a yearning for it. If you have a yearning for ice cream, it probably means that you've gone a long time without any and are almost desperate to taste some. The Old English root of the words yearn and yearning is georn, which means "eager or desirous."