Definition: An element is any thing.
That’s right, that’s all. An element is so basic that it hardly warrants a definition. These are undoubtedly the most fundamental objects in mathematics, and arguably the most fundamental objects in all of human thought. These are any kind of thing at all. They make up sets and get “sent” to other sets by functions. Most textbooks don’t even attempt to fully define an element, and so I’ll simply end this here since the intuition for an element as a “thing” is all that is needed.