Meaning of finite, countably infinite, infinite? Even after several attempts I could not find the motivation behind the finite, countable  and infinite. Is there a simple way to look them differently? I have read the wikipedia definition several times.
 A: We say that two sets have the same cardinality if there is a bijection between them. This is the foremost key to understanding what these definitions mean, because they are, in essence, saying something about cardinality of sets. 
I will also assume that you have some intuitive understanding of what is a natural number.


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*$A$ is a finite set if there is a natural number $n$, such that there is a bijection between $A$ and $\{0,\ldots,n-1\}$. Namely, a finite set is a set whose size corresponds to a natural number in the most naive and intuitive sense that you can imagine. The empty set has $0$ elements, $\{x\}$ has $1$ element, and so on.

*$A$ is infinite if it is not finite. As simple as that. I remember seeing this definition for the first time, and I chuckled, because it seemed like a strange definition. But with time I grew to appreciate it as a very correct and "on the nose" definition.

*$A$ is countably infinite if there is a bijection between $A$ and the set of all natural numbers, $\Bbb N$. In particular $A$ is infinite, since there are infinitely many natural numbers.
There are many important basic theorems, which this answer is too short to cover. I suggest you open some basic book about naive set theory to understand the connection between these notions better.
The big difference is between infinite and countably infinite, of course. But one of the deeper theorems of set theory, Cantor's theorem, says that there is no maximal cardinality. So there is always a larger set. In particular, there are infinite sets which are not countably infinite. $\Bbb R$ is the first example of such set. On the other hand, $\Bbb Q$ is countable.
A: To put it simply:


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*$S$ is finite: you can write $S = \{x_1,x_2,\cdots, x_n  \}$ for some $n \in \mathbb{N}$.

*$S$ is countably infinite: you can write $S = \{x_1,x_2,\cdots, x_n,\cdots  \}$.

*$S$ is uncountably infinite: you can't list all the elements in $S$.
A: Countable means "at most the size of $\Bbb{N}$", so it is either finite or the size of $\Bbb{N}$. 
Infinite means "at least the size of $\Bbb{N}$", but it might be bigger.
A: Let us look at these terms in light of sets. Sets are collections of objects.
Finite: Finite sets are those that have a maximal and a minimal elements; that is, there is an end to the number of its members. For example, the set $\{1,2,3,4,5\}$ is finite.
Countably infinite: Countably infinite sets are those that for example have no maximal or minimal element. These are the sets that you could assign a number to each member. The set of natural numbers $\{1,2,3,\ldots\}$ is a countably infinite set.
(uncountably) Infinite: These are the sets that are a little tricky, and are better defined by example. These sets can indeed have a maximal and minimal element, but still have an infinity of members in between. The set of real numbers is a uncountably infinite set. Why? Well, let us look at a subset of the real numbers, the interval between $0$ and $1$. Obviously, there are an infinite amount of numbers in here. Not convinced? Pick any number in that interval, call it $n$. Then, $n/m$ is also a number in that set, provided that $m\ge1$. Since any natural number satisfies this condition, and said set is infinite, there are, for every number in that interval, an infinite amount of numbers.
What about assigning a number to each member, to show it is countably infinite? Well, this is impossible, as you cannot number each member, as, when you were done (if you were done) numbering this set, there would still remain an infinite amount of members to be numbered!
