What is the difference between Modulus, Absolute value and Modulo?
They mean differently.
Absolute value of $x = |x|$ and is equal to $x$ if $x \geq 0$ or is equal to $-x$ if $x < 0$.
Modulo, usually refers to the type of arithmetic called modulo arithmetic.
For example, because $13 = 4\times 3 + 1$, we write $13\ \equiv\ 1\ (\textrm{mod}\ 3)$ ($13$ is congruent to $1$ modulo $3$).
Modulus refers to the magnitude/length of a vector.
Added
How about “An introduction to the theory of Numbers – by Niven Zuckerman” and “Pure Mathematics I & II by F. Gerrish”?
Those names have been commonly used by others and sometimes even interchangeably but, in the books mentioned above, they are clearly and distinctly defined.
The only confusion comes from the “$|…|$” sign, which has been used both for the absolute value of a number and also as the modulus of a vector. Therefore, some used the “$|| … ||$” for the latter to make the meaning distinct. Some don’t even bother when the context is clear or when the readers should be able to distinguish their difference.
It is an unfortunate fact that mathematical terminology has developed in a haphazard way, and often the same word will be given completely different meanings in different areas of mathematics (sometimes even in the same area, which is worse!). The most extreme example, I think, is "normal".
See http://jeff560.tripod.com/m.html for some history of the many uses of "modulus" and its variants.
If $z = a + b \imath$ then $|z| = \sqrt{a^2 + b^2}$ is called its modulus. If you work with both real and complex numbers frequently it is common to misspeak and call the absolute value $|x|$ of a real number $x$ its modulus.