The set of countable ordinals as as least uncountable as can be. It is exactly the ordinal $\omega_1$ and has size $\aleph_1$. The real numbers, on the other hand, are provably uncountable, but can have an arbitrarily large cardinality. So it would be at least of size $\aleph_1$ (in $\sf ZFC$, anyway).
As for the term "see", if the above explanation is not sufficient and you are looking for a geometric visualization then you're asking for trouble. While we can easily understand the real numbers as a "line" the ordinals are more difficult because they have limit points, but $\omega_1$ has so many limit points that just drawing the limit points, or the limit of limit points, or the limit of limit of limit points, and so on, all that would amount to the same drawing.
So it becomes hard to visualize $\omega_1$, and these visualizations never help us understand the cardinality of anything anyway. But wait, it gets worse.
Given $M$ a [countable] transitive model of set theory, and $\alpha$ is an uncountable ordinal in $M$, there is a generic extension of $M$ where $\alpha$ is in fact countable internally. The fact it is a generic extension tells us that there are no new ordinals, so without adding new ordinals we made the ordinal countable.
So it's hard to get an accurate image of $\omega_1$. My rule of thumb is that whatever you think it is, it's larger, much larger. But still quite small -- being the least uncountable ordinal. And $\Bbb R$ is at least as big.