Something that's everlasting is eternal; it lasts forever, or seems like it does. Many anti-war activists dream of everlasting peace.
The adjective everlasting can literally mean exactly what it sounds like, but it's frequently used colloquially to exaggerate. You might compliment your friend's speech by saying, "You have my everlasting respect," or describe a lovely day by saying, "It's been like an everlasting dream." In a religious context, the word is used quite literally: "God promises everlasting life."