Diffusion is one of those really, really important concepts which is impossible to understand first off, but when you've got your head round it, it's about as simple as things come! And it's used in so many aspects of science, that understanding it is an incredible help.
Put most simply, diffusion is the even spreading of something - that is, if you have a huge amount of something in one small area, then the diffusion of that particular substance would mean that it would be spreading out to occupy a larger area. When a small drop of ink falls into a beaker of water, the ink diffuses out as it fills the water. When someone breaks wind, the smell diffuses through the room as it spreads out.
The strict definition of diffusion is the movement of particles from an area of high density to an area of low density, or the movement of a substance down its concentration gradient. This is basically just a posh way of saying that things move from where there's lots of it to where there's not much of it. It's almost like a sympathy thing - if one particle sees another particle feeling lonely, it goes to look after it until everything's equal. Of course, particles don't think, so it's not really like that, but in a sense it helps to think about it like that. Everything needs to be spread out evenly, and so if there's more of something on one side, the process of diffusion makes particles from one side move to the other until they're equal.
If we go a little deeper, we can see why this works. Let's take a gas. The particles of a gas will be moving about anyway, bumping into each other, and into the walls of whatever container they're in. So particles will be moving around randomly. And lets say that in each case, a fifth of the particles cross to the other side.
So if there are 100 particles on the left side, and only 10 on the right, then 20 particles will move from left to right, but only 2 from right to left. So now there are 82 on the left, and 28 on the right. Then about 16 move from left to right, and 6 move from right to left, leaving 72 on the left, and 38 on the right. So eventually the gap between left and right is narrowed, and the two even out.
This is an important consideration in relation to diffusion. It's not simply a case of movement in one direction. Particles move in both directions, but because one side has more particles to start with, they are more likely to move in one direction than the other. When both sides have the same number of particles, they are just as likely to move one way as the other way, so there will be no change in amounts - both sides will stay equal.
Clearly it takes a certain amount of time for the two to even up. This amount of time varies depending upon certain factors. If there is a huge difference in concentration (i.e. lots of one, and not much of the other), then a fifth of the larger one is going to make a big difference to the smaller one. At the beginning of the example above, when 20 moved from the larger concentration to the lower concentration, it made a big difference. However, as they got closer, the rate slowed (it took longer for the two to even out). So difference in concentration has an important effect on the rate at which diffusion happens.
Diffusion can also happen across a surface. Especially in biology, we talk about things diffusing across a membrane. Obviously because there is this barrier in the way, the process will be slower, but certain substances can still get across a membrane because it's not a completely solid barrier - there are very tiny gaps between the phospholipids of the membrane. So whatever is diffusing through has to squeeze between these gaps - and it takes longer, because they don't so much squeeze through, as fly towards it in the hope that they find a gap. Any particles which don't find a gap will just bounce back off, so its all a very random process. By the very nature of the process, however, they will eventually get through so that there is as high a concentration on one side as there is on the other.
With gases, we tend to refer to pressure rather than concentration - and most specifically, partial pressures. If there are two gases, one on either side, then they will mix with each other - but again, there should be as much of one gas on one side as there is on the other. Once the gas has evened out, the partial pressure of that gas should be the same on both sides of the membrane, assuming the gas is dissolved in the same substance.
If the gas is going from air, through a membrane, into air, then it's nice and simple. However, if it's going from air, through a membrane, into (for example) water, then the gas isn't necessarily as happy in the water. It will dissolve in the water less easily. But the 'difference in concentration' rule still applies; if there is a larger difference in concentration between the gas in the air and the gas in the water, then the rate (or speed) of diffusion will happen much quicker. Unfortunately, because you're not comparing like with like (i.e. air and water, not air and air), when the gas 'evens out', it won't necessarily be the same on both sides.
Ultimately what you're looking for is a state of equilibrium, where there is as much of the substance going from left to right as there is going from right to left.
A rather clever man named Adolf Eugene Fick (1831-1879) did some experiments which revealed important information about diffusion. His 'law' showed that the rate of diffusion doesn't just depend on difference in concentration (or difference in partial pressures), but also on the (surface) area of the membrane, and on the thickness of the membrane. We've already established that if there's a huge difference, it will even out more easily. However, if there is a larger area to work with, then it will be easier to find a gap - the substance will be more likely to go through. And, perhaps obviously, if there is a greater thickness to get through, the chances of getting through are slimmer. Only he said it in a clever way...
Unfortunately osmosis is where the concept of diffusion gets even more confusing. Rather than the simple process of one chemical evening itself out, osmosis relates to a situation where something is dissolved in something else. It is initially rather important that you understand the difference between a solvent and a solute - a solute being the thing which is dissolved, and the solvent being the thing which it is dissolved in. If you make a cup of tea and decide to have some sugar in it, then the drink of tea is the solvent because you are dissolving into the tea, and the sugar is the solute, because it is dissolving into it.
The definition of osmosis is very difficult to understand, but once it is explained, it makes a lot more sense. Put very simply, osmosis - when refering to water - is effectively the diffusion of water molecules. The strict definition is the diffusion of a solvent, through a partially permeable membrane, from an area of low concentration to an area of high concentration.
The partially permeable membrane bit is easy - simply a biological membrane, the phospholipid bilayer you find around every cell. But the rest is not so simple. When we look at the section on diffusion above, we see that things diffuse from where there is more to where there is less - so why do we say 'low concentration to high concentration'? Well, there's two ways of thinking about this.
If you understand the concept of equilibria, think about the system trying to establish an equilibrium. If you have one cup of weak tea and one cup of strong tea, and you have to add water to one of them to make them the same strength, then you dilute the cup of strong tea. In the same way, if you have a strong solution (i.e. high concentration) and a weak solution (i.e. low concentration), the only way to get them in equilibrium is to move some of the solvent from the weak solution to the strong solution. If you do it the other way round (i.e. move the solvent from the strong solution to the weak solution) then the strong solution will get stronger, and the weak solution will get weaker, and they will not even out.
The other way of thinking about it imagines that you are talking about the solvent being 'free'. For the purposes of this, the solvent will be water, and the solute is sugar - you're dissolving sugar in water. If you have lots of sugar dissolved in the water, then the water will all be occupied - there isn't much 'free' because it's all looking after the sugar. On the other side, there isn't so much sugar, so there's more free water. In order to even this out, some of the 'free' water can move to the other side, through this membrane, so that there's less free water on the weak side and more free on the other.
Talking about the solvent being 'free' isn't entirely accurate, but it may help to understand what's going on. Ultimately it's always trying to make things fair - to even things out. The process of osmosis is very, very important for biological systems, because volume can be regulated by osmosis. Normally you'd assume that if one thing had a large volume and the other had a small volume, water would tend to go towards the small volume to even things out.
But, if there were lots of salt dissolved in the large volume, then in order to even out the concentration, you need to add more solvent (e.g. you need to add more water), and this will come (by osmosis) from the smaller volume. The animations display this. As you can see from the first animation, water molecules move from the cell on the left to the cell on the right through the semipermeable membranes between. This makes the bigger cell even bigger. The second animation shows why this happens - the result of the increase in water in the bigger cell means that, although there is a greater difference in size, the concentration of the yellow particle is now the same in both.
It's a difficult topic, but when you understand it, many of the ways in which the body regulates blood volume and cell volume become clear. The body is a fantastically designed machine, but it's also very, very complicated!