Whole Numbers and Integers
Whole Numbers
Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on)
No Fractions!
Counting Numbers
Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero. So they are 1, 2, 3, 4, 5, … (and so on).
Natural Numbers
"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.
Integers
Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed!
So, integers can be negative {1, 2,3, 4, 5, … }, positive {1, 2, 3, 4, 5, … }, or zero {0}
We can put that all together like this:
Integers = { ..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ... }
Confusing
Just to be confusing, some people say that whole numbers can also be negative, so that would make them exactly the same as integers. And sometimes people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing!
My Standard
I must admit that sometimes I say "negative whole number", but usually I stick to:
Numbers

Name

{ 0, 1, 2, 3, 4, 5, … }

Whole Numbers

{ 1, 2, 3, 4, 5, … }

Counting Numbers

{ ... 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, … }

Integers

But nobody disagrees on the definition of an integer, so when in doubt say "integer", and if you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent. Like this (note: zero is neither positive nor negative):
 Integers = { ..., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ... }
 Negative Integers = { ..., 5, 4, 3, 2, 1 }
 Positive Integers = { 1, 2, 3, 4, 5, ... }
 NonNegative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?)