Before I describe the physics, let me tell you that you already know most of this. The molecules in the earth's atmosphere scatter light from the sun. Because of the quantum mechanical properties of the molecules, they scatter blue light more efficiently than red light. Now the sun light you see in the sky away from the sun, you can see only because it is scattered toward you - and so this makes the sky blue. At sun set or sun rise, the sun light passes through a lot more atmosphere than at midday. The increased path length causes the strong reddening of the sun. This is partly due to scattering, partly due to absorption on little dust grains. For example, volcanic eruptions spew lots of dust grains into the atmosphere, which cause amazingly red sun sets. So dust particles also cause reddening. The same process of scattering and absorption also occur in the ISM.
Scattering changes the direction of the incoming photon, but not it's energy. This type of reflection can also induce polarisation of the light, if the dust grains are aligned in a given direction, for example due to a magnetic field. You know that scattering can cause polarisation: expensive sunshades are polarised, so as to block incoming polarised light reflected from the ocean, or road surface for example. So dust grains may scatter some of the light coming from a distant star out of the line-of-sight, thereby reducing the amount you detect, and hence the apparent luminosity of the star. Given the typical size of interstellar dust grains, blue light is scattered more than red light, and hence scattering also leads to reddening. Now if you look at a dust cloud, you may actually detect some of the light scattered off dust from a nearby bright star for example. And so such a reflection nebulae is typically blue (so for the same reason that the sky is blue, except it's scattering by dust (for the reflection nebula) vs by molecules (for the earth's atmosphere)).
When a dust grain absorbs a photon, it can sometimes undergo a change in structure, for example a molecular bond will be broken, or an atom or molecule gets into an excited state. The energy of the photon is converted into another form of energy, and is lost to us: the photon is absorbed. So one effect of absorption is to decrease the amount of light we detect from the star. Very often, the dust grain will emit other photons, but at much lower wavelengths, typically in the infrared. For example in star forming regions where the young stars produce large amounts of dust, and so, even though these stars may be very bright, we cannot see them at visual wavelengths because of the large amount of absorption by dust. Conversely, these regions are very bright in the infrared, because of the reprocessing. If the wavelength of the light is much larger than the size of the dust grains, then absorption is much reduced. And so you understand that the newly planned space-telescope which will work in the infrared, will teach us a lot about star forming regions.4.1
The scattering and absorption by dust grains may seem to be just a nuisance for astronomers, given that they decrease the amount of star light we observe. But because the absorbing properties depend on the size and composition of the dust, we can learn about dust properties by studying their effect on star light. And so this is how we infer the typical size and composition of the grains. Although the amount of absorption typically decreases with increasing wavelength, there is much more absorption at some specific resonances, which teach us that some grains have graphite in them (graphite, like diamond, is a phase of carbon), and also silicates (we detect the resonance at the energy of the Si-O chemical bond).