Mole is the SI unit used to measure how many molecules or atoms there are. One mole is around 600 sextillion molecules. Scientists use this number because 1 gram of hydrogen is around 1 mole of atoms.

The exact value of one mole is . This number comes from experiments with carbon because it's easy to work with.

 is also called Avogadro's number, which was named after the person who invented it.

Anything can be measured in moles, but it is not practical for most tasks because the value is so massive. For example, one mole of grapefruits would be as big as the earth. 

Because different molecules and atoms do not have the same mass, one mole of one thing does not weigh the same as one mole of something else. Atoms and molecule mass is measured in amu. One amu is equal to one gram per mole. This means that if an atom has a mass of one amu, one mole of this atom weighs one gram.

Mathematics with the mole 
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 methane molecule is made from one carbon atom and four hydrogen atoms. Carbon has a mass of 12.011 u and hydrogen has a mass of 1.008 u. This means that the mass of one methane molecule is 12.011 u + (4 × 1.008u), or 16.043 u. This means that one mole of methane has a mass of 16.043 grams.

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. Other common units include a dozen, meaning  12, and a score, meaning 20. Similarly, a mole refers to a specific quantity-- its distinguishing feature is that its number is far larger than other common units. Such units are typically invented when existing units can not describe something easily enough. Chemical reactions typically take place between molecules of varying weights, meaning measurements of mass (such as grams) can be misleading when compared the reactions of individual molecules. On the other hand, using the absolute number of atoms/molecules/ions would also be confusing, as the massive numbers involved would make it all too easy to misplace a value or drop a digit. As such, working in moles allows scientists to refer to a specific quantity of molecules or atoms without resorting to excessively large numbers.

Related units
The SI units for molar concentration are mol/m3. However, most chemical writing uses mol/dm3, or mol dm-3, which is the same as mol/L. These units are often written with a capital letter M (pronounced "molar"), sometimes preceded by an SI prefix, for example, millimoles per litre (mmol/L) or millimolar (mM), micromoles/litre (µmol/L) or micromolar (µM), or nanomoles/L (nmol/L) or nanomolar (nM).

The absolute yield of a chemical reaction mostly stated in moles (called the "molar yield").

References 

Chemistry
SI units