Intuitively speaking, if $A$ is countable, you have a way to list out the items in $A$. If $A$ is uncountable, you will not have such ways, and if you try to make a list of $A$, the list must have some elements missing(between the elements in the list).

You can take $\mathbb Q$ and $\mathbb R$ for corresponding examples.

A denumerable set is in one-to-one correspondence with the set of natural numbers. Such sets may also be called countably infinite, see

Such a set has the form $A = \{a_0,a_1,a_2,\ldots\}$ and so can be tabulated by the natural numbers (indices). Examples are the natural numbers, the even or odd natural numbers, the integers and the rational numbers.

Basic non-denumerable sets are the real numbers or the power set of the set of natural numbers which are equipotent.

For a finite set $A$ the size of $A$ or $card(A)$ is the number of elements in $A$. Now, what's the size of infinite sets? Or first: what's the definition of infinite set?
A set $A$ is infinite if there exist a $1-1$ and onto map $f:A\to B \varsubsetneqq A$!
Since we cannot set a number as size of an infinite set, instead we use $card(A)$ as size of set $A$; and we can compare cardinality of sets. We say $card(A)=card(B)$ if and only if there exist a $1-1$ map $f$ such that $f:A\to B$
In many cases finding such mapping is extremely hard. so instead we use this theorem:
$card(A)=card(B)$ if and only if there exist two $1-1$ maps $f$ and $g$ such that $f:A\to B$ and $g:B\to A$. Intuition of $f$ and $g$ is (sort of) $A\subseteq B$ and $B\subseteq A$. In many cases finding such mappings is so easy.
Now, an infinite set $A$ is countable infinite if $card(A)=card(\mathbb{N})$ else it is uncountable infinite.
As some examples, $\mathbb{N,Z,Q}$ are countable infinite and $\mathbb{R},\mathbb{C}, \mathbb{R}^n, P(\mathbb{R})$(powerset) are uncountable infinite.
For a countable infinite set $A$, let $f:A\to \mathbb{N}$ be $1-1$ and onto, so we can write elements of $A$ as $\{a_1=f^{-1}(1),a_2=f^{-1}(2),...\}$ so intuitively we can count elements of $A$ exactly like $\mathbb{N}$ as first element, second element and so on.
But for sets like $\mathbb{R}$ we can't count elements like $\mathbb{N}$, witch means $card(\mathbb{R})$ is much greater than $card(\mathbb{N})$.