A set with uncountable number of elements is called an infinite set.
Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set?
As far i know $\Bbb N$ is "countably" infinite or denumerable.
That is, i can't say $\Bbb N$ is "uncountable".
Then how is $\Bbb N$ an infinite set?
On
Finite set is a set that for some natural number $k$, there is a bijection between the set and $\{1,\ldots,k\}$.
An infinite set is a set which is not finite. $\Bbb N$ is not finite, so it is infinite.
Countably infinite set is a set that has a bijection between itself and $\Bbb N$. An uncountable set is a[n infinite] set which is not countable. So an infinite set can be countably infinite or uncountable.
(If it bothers you that the definition of finiteness appeals to the natural numbers there are set theoretical definitions which makes no such appeal. For example Tarski's definition $A$ is finite if and only if every non-empty $U\subseteq\mathcal P(A)$ has a $\subseteq$-minimal element; or Dedekind's definition $A$ is finite if and only if every injection from $A$ to itself is a bijection.
Interestingly the equivalence between the definition in my first line, and Tarski's requires no use of the axiom of choice, whereas the equivalence with Dedekind's definition does require us to use the axiom of choice.)
On
Finite/infinite and countable/uncountable are in different level when we try to describe "how large" a set is. $$\begin{equation} "How\ large"\ a\ Set\ is \left\{ \begin{array}{ll} finite&\\ infinite& \left\{ \begin{array}{ll} countable&\\ uncountable& \end{array} \right. \end{array} \right. \end{equation}$$ Which means, if we try to describe how large a set is, firstly, we should see whether it is finite or infinite.
For finite sets, we can easily say which one is larger by comparing how many elements they contained. However, for the sets with infinite elements, one may not compare them by the usual way.
But somehow, we still want to know which one is larger. Then, we use countable and uncountable to describe them.
For example, $\mathbb{N}$ and $\mathbb{R}$ are both infinite sets, but it is obvious that $\mathbb{R}$ contains more than $\mathbb{N}$. Since there are much real numbers between any two integers, but this doesn't hold for integers, like between 2.1 and 2.2.
In fact, we use "Cardinality" to say how large a set is, and the Cardinality of $\mathbb{N}$ is denoted by $\aleph_0$, and $\mathbb{R}$ is equal to $\aleph_1$, if you believe continuum hypothesis.
Sets can be grouped into three sizes: finite, countably infinite, and uncountably infinite. The natural numbers are countably infinite, and thus infinite. In fact, the very definition of being countably infinite is based on the natural numbers: a set is countably infinite if and only if there is a bijection between that set and the natural numbers.