A mole looks like this:
But as usual, the chemists have to take a perfectly nice animal and pave over it with some non-related term. Sadly, the latter is probably the one you're asking about...
In the context of chemistry then, a mole (usually bizarrely abbreviated to 'mol') is simply a certain number of atoms or molecules. In fact, it's a very particular number, Avagadro's number, or roughly 6.02 x 1023, of those atoms or molecules.
Fortunately, in medicine you don't need to remember this unwieldly number. Wikipedia describes a mole as being an analogue to a 'dozen', and that's a nice way of putting it - a mole is just a word that indicates that there are a certain number of <whatevers> present, except that unlike a dozen, the number isn't 12; it's Avagadro's number.
For instance, in the solution I gave to a patient yesterday, there were 158 millimoles (158 mmol) of sodium per litre. If I wanted to, I could work out exactly how many sodium molecules there were in the litre, but the number would be cumbersome and entirely futile to deduce. So, instead, we use a convenient term that converts this number into a far more manageable and user-friendly number.
There's an analogy here to astronomy. In that field, the distances measured are absolutely enormous, so instead of clumsily writing the distance between the sun and proxima centuri in metres, the astronomers use light years, a different system of measuring distance that allows distances to be written and conceptualised in more manageable terms. Suddenly, proxima centuri's distance is converted from roughly 39 924 282 590 234 214 738 126 847 metres to 4.22 light years. So it is with the number of sodium atoms in my patient's drip fluid.