Just to add: a semi-permeable membrane acts like a very fine netting and is usually considered to only allow small molecules through (such as water), while blocking large molecules (such as sugar). If you divided a container in half with semi-permeable material, part-filled one half of the container with weak sugar / water solution and part-filled the other with strong sugar / water solution, then osmosis could be seen.
The water molecules, being all that can fit through the tiny holes, are constantly passing through it in both directions, just as they would constantly move if there was no membrane. However, slightly more water enters the strong sugar solution half than leaves it. Over time this results in the side that originally had most sugar, to also have the most water (it drew the water through the membrane). The special thing about how much extra water it has 'stolen' is that the concentrations of both halves end up the same. For example, at the start, the concentrations of sugar-to-water molecules might have been 4:1 and 10:1. Assuming that both halves contained the same amount of water, at the end the concentrations would be 7:1 and 7:1, and one half would have more water in it than the other. The sugar, of course, never changed sides ' just the water.
I'll leave it to someone else to explain the exact chemistry of why concentrated solutions attract water, because I have no idea!