We've already answered the question, "What is a mole?". Now it's time to steer the session more distinctly towards biology and medicine: what is an osmole?
Osmosis depends on the number of solute particles in a solution. In calculating the number of particles, it is important to consider that a salt (like NaCl - table salt) may well dissociate in the solution. If it does, then you have to consider the number of moles of Na+ and the number of moles of Cl- when thinking about osmosis. If you just considered the number of moles of NaCl, you would underestimate the osmotic force. In other words, 1 mol of NaCl gives 2 moles of osmotically-acting particles.
Note that many molecules don't give you this sort of headache, since they don't dissociate. 1 mol of, say, lactose still only gives 1 mol of osmotically-acting particles.
To encapsulate this in a concept, biochemists often talk of osmoles, which are the number of moles of osmotically-acting particles in a solution.
Putting that altogether, you could say that 1 mol of NaCl contains 2 osmoles.
Usually, a full osmole is too great a number for convenient use with reference to the human body, and so milliosmoles (mOsm) is often used instead - one milliosmole being one thousandth of an osmole of course.
(Whew! That was dry, wasn't it? Sorry...)