Elaboration of infinite, finite and enumerable definition I am starting to learn some of the basic concepts of math. The concept I am learning now is infinite, finite, and denumerable. I am having trouble understanding the book's definiton. I am hoping if someone can elaborate what it exactly means. The book states:

A set $B$ is finite if it is empty or if there is a one-one function
  with domain $B$ and range in an initial segment of $\mathbb{N}$. If
  there is no such function, the set is infinite. If there is a one-one
  function with domain $B$ and range equal to all of $\mathbb{N}$, then
  the set $B$ is denumerable (or enumerable). If a set is either finite
  or denumerable, it is said to be countable.

 A: First is the definition of a finite set;
"one-one function with domain B  and range in an initial segment of $\mathbb{N} $" just means that you can write the elements of your set in a list, one entry on each line, using n lines only, where n is  natural number. 
So if your set is {2,3,4}, you write 2 on the first line, 3 on the second and 4 on the third line. Now 2 becomes the first element in your set even though it's the second natural number, because we put it on the first line. 
You don't ever skip lines in the construction of the list so the image will always be an intial segment of $\mathbb{N}$
An infinite set;
"If there is no such function, the set is infinite" If you can't write your set out on n lines, where n is a natural number then it is said to be infinite. Something like the decimal expansion of $\pi$ is said to be infinite.
A countable set;
"a one-one function with domain B  and range equal to all of N , then the set B  is denumerable" This means you can create a function that will write out all the elements of your set in a list. You can talk about your set having a first element, a second element and so on even if it has infinitely many elements. Think of the even numbers $2\mathbb{N}$ = {2,4,6...}. Our bijection is n/2. Take any number in $2\mathbb{N}$, half it, and whatever number you get you put it on that line in your list.
For an example of a set that is uncountable see the proof that $\mathbb{R}$ isn't countable, where one shows you can't write out all its elements in a list(or give a function to do so).
A: If you have a one-to-one function $f\colon A \to B$ with range $B$ you have what is also called a bijection between $A$ and $B$. Intuitively we would say that in that case the two sets have the same quantity of elements. The definition of your book says that a set is finite if it has a bijection with one of the set:
$$
  \{\}, \{1\}, \{1,2\}, \{1,2,3\}, \{1,2,3,4\}...
$$
For example the set $\{2,4,6,8\}$ has a bijection with $\{1,2,3,4\}$ and hence is finite. If a set has a bijection with $\mathbb N$ then it is denumerable. This means that you can count all the elements of your set using natural numbers, but you will never stop. 
