"A mole of a substance may be defined as the molecular weight of that substance expressed in grams. The molecular weight (m wt) of glucose is 180 so 180g of glucose = 1 mole. The m wt of NaCl is about 58 so 58g of NaCl = 1 mole. If 58g of NaCl are placed into a beaker and water added to a volume of 1l then the result will be a 1 molar solution. Which may be shown as 1mol/l or 1mol l-1 or 1M. Something that you may come across in textbooks are molal solutions. A molal solution is, for all practical purposes identical to a molar solution. The only difference is that instead of placing the salt in a beaker and adding water to a volume of 1 l, the salt is added to a litre of water. There can only be a difference in the two solutions if less than 1 l of water is required to bring the final volume (of salt and water) up to 1 l. This is the sort of nit-picking that chemists love and biologists ignore. Molal solutions may also be written as mol/l or mol l-1 or m. Don't even get me started on normal solutions"
Moles are used as a mass unit by biologists (and chemists) in preference to grams because a mole of any substance contains the same number of particles as a mole of any other substance. This greatly simplifies calculations that deal with chemical reactions (including biologically relevant reactions such as buffering) and osmotic efffects. The number of particles, atoms or molecules in a mole of any substance is given by Avagadro's number which is about 602214199000000000000000 or 6 x 1023 (mol-1)
Osmolarity is a measure of the osmotic pressure exerted by a solution across a perfect semi-permeable membrane (one which allows free passage of water and completely prevents movement of solute) compared to pure water. Osmolarity is dependent on the number of particles in solution but independent of the nature of the particles. For example, 1 mole of glucose dissolved in 1 litre of water has an osmolarity of 1 osmole (osm) /l. If 1 mole of another sugar, such as sucrose were added to the same litre of water, the osmolarity would be 2 osm/l. It doesn't matter that the solution contains 1 mole of glucose and 1 mole of sucrose. If 1 mole of NaCl were dissolved in 1 litre of water it would produce a 1 mol/l NaCl solution with an osmolarity 2 osm/l because NaCl dissociates into Na+ and Cl- (two particles) in solution. This is true of all compounds that dissociate in solution. Na2SO4, which dissociates into Na+, Na+ and SO42-, to give 3 particles per molecule produces 3osm/l for every mole dissolved in 1 litre.
If two solutions contain the same number of particles they may be said to be iso-osmotic (isosmotic) with respect to each other. If one solution has a greater osmolarity than another solution it is hyperosmotic with respect to the weaker solution. If one solution has a lower osmolarity than another solution then it is hypo-osmotic (hyposmotic) with respect to the stronger solution. Iso, hyper and hypo osmolarity should always be stated with respect to another solution. For example, a 1 mol/l NaCl solution is hyperosmotic with respect to 1 mol/l glucose solution.
Tonicity is nearly the same as osmolarity. For substance that cannot cross cell membranes, tonicity is practically identical to osmolarity. Tonicity is a measure of the osmotic pressure that a substance can exert across a cell membrane, compared to blood plasma. Plasma has an osmolarity of about 0.3 osm/l, therefore a 0.15 mol/l NaCl solution may be said to be isotonic with plasma (Assuming that neither Na+ nor Cl- can cross cell membranes, which is nearly true). If a substance can cross a plasma membrane, then it cannot exert an osmotic pressure across that membrane. The solute will equilibrate across the membrane instead of forcing water to move. Urea behaves like this, so a 0.3 mol/l urea solution may be said to be iso-osmotic with plasma but it is not isotonic.
In The Real World...
Absolutes are very hard to find in biological systems, there are probably very few small molecules that cannot cross a plasma membrane at all. However, what is important is the rate at which they do so, compared to water. Even if a membrane does allow a little Na+ and Cl- across, the rate at which the NaCl concentration would equilibrate is much, much slower than the rate at which water moves by osmosis. The problem with urea is that it gets across cell membranes nearly as fast as water does.
All of the above are perfectly true only in a simplified universe in which all solvents behave as perfect liquids, all ions dissociate completely and no ions or other particles interfere with each other in any way. Real life is, as usual, more complex. For example, if you were to calculate the osmolarity of plasma by adding up the concentrations of all the anions and cations in solution, you would get a value slightly above 0.3 osm/l (300mosm/l). If the osmolarity of plasma is measured directly by freezing point depression the value comes out to be about 0.29 osm/l. Ho hum.