The measurement of Avogadro's number was refined in 2011 to 6.02214078×1023 ± 1.8E-7×1023. 
We use this number because it is the number of
The number does not lend itself to easy expression in words. The nearest "casual" number is one million-million-million-million, which is 1024.
Because different molecules and atoms do not have the same
Moles = mass (g) / Relative mass (grams per mole) Example: How many moles are there in 20 grams of hydrogen? A value of 1 can be used for hydrogen's relative mass, although the correct value is slightly larger. So: moles = mass/relative mass = 20/1 = 20 moles.
Moles = concentration (mol/dm3) x volume (dm3) Example: How many moles are there in 100cm3 of 0.1M H2SO4? 1 dm3 is the same as 1000 cm3, so the value in cubic centimetres needs to be divided by 1000. 100/1000 x 0.1 = 0.01 moles.
A mole can be thought of as two bags of different sized balls. One bag contains 3 tennis balls and the other 3 footballs. There is the same number of balls in both bags but the mass of the footballs is much larger. It is a different way to measure things. Moles measure the number of particles, not the mass. So both bags contain three moles.
A mole is simply a unit of the number of things. Units are invented when existing units can not describe something well enough. Chemical reactions often take place at levels where using grams wouldn't make sense, yet using absolute numbers of atoms/molecules/ions would be confusing, too.