An ELI5 answer: Think about "talking a walk" along the Euclidean Line -- the real-number line, if you like. Face in one direction, and then start taking steps.

An ELI5 answer: Think about "talking a walk" along the Euclidean Line -- the real-number line, if you like. Face in one direction, and then start taking steps. Stepping Example #1. Start at 0 and face toward the positive numbers. Step to 1. Step to 2. Step to 3. Step to 4. Etc. Each time you take a step, just a bit more of the line is behind you. And if I pick a point on the line, then no matter which point I pick, this way of stepping will eventually put that point behind you. Stepping Example #2. Start at 0 and face toward the positive numbers. Step to 1/2. Step to 2/3. Step to 3/4. Step to 4/5. Etc. Again, each time you take a step, just a bit more of the line is behind you. But this time, I can find points -- 6, for example, or 300 million -- that will never be behind you. (There is a technical term for the first kind of stepping. We say that the set of points you step on, is cofinal in the order on the Euclidean Line. The second kind of stepping does not have this property.) One of the curious properties of the Long Line is, first, that "locally", it is exactly like the Euclidean Line. Stand anywhere on the Long Line and look around. "Looks like I'm on the Euclidean Line," you say. But, second, every possible way of stepping ends up like Example #2 above. No matter what steps you choose to take, I can find a point -- lots of points, in fact -- that will never be behind you. And that is why we say it is "long". (Formally, the Long Line is locally homeomorphic to the Euclidean Line, but the order on the Long Line admits no countable cofinal subset.)