The mole is a specific quantity, 6.02x10^23. A mole of anything will contain this number of particles, just as a dozen of anything will contain 12. The mass of a mole of particles depends on the mass of the individual particle, just as a dozen golf balls doesn't have the same mass as a dozen ping pong balls.
The molar mass of a substance is the mass of one mole of particles in grams. For example, since the molar mass of carbon is 12 g/mol, one mole of carbon atoms has a mass of 12 g. H2O has a molar mass of 18 g/mol, (2xH) + O. One mole of water molecules therefore has a mass of 18g.
The molar mass is used as a conversion factor to convert grams to moles and moles to grams, as follows:
moles x molar mass = grams
grams/molar mass = moles