Something abiding sticks around, lasting a long time. Abiding is usually used with feelings and memories — as in your abiding love for Elmo.
Abiding comes from the Old English abidan, gebidan meaning "remain, wait, delay, remain behind." It's a word that's typically used to describe a feeling or memory that remains behind or lingers in your mind a while. You can have an abiding faith in God, or an abiding respect for war veterans or an abiding passion. "Law-abiding citizens" are those people who stick strictly to the law. They always use the crosswalk and pay their taxes on time.