Vapor is something that forms over any liquid and does not require the liquid to be above its boiling temperature. The atmosphere is a great example; it contains about 37 billion gallons of water in the form of vapor, but the temperature of (most of) the atmosphere is well below the boiling point of water. The goal of a vaporizer is simply to increase the rate of this process.
To understand how it does this you need to know a little bit about vapor pressure. A liquid at a given temperature will have some of its molecules which are moving fast enough to escape the surface of the liquid and end up in the vapor phase above the liquid. If you put the liquid in a sealed container, this process will continue until the partial pressure of the vapor reaches the vapor pressure of the liquid. What is important about this process is that it is only the pressure of the substance itself (i.e. partial pressure) which matters, even if the rest of the air above the substance is at a higher pressure than the vapor pressure, the liquid will still vaporize until its partial pressure reaches the vapor pressure. The vapor pressure increases with increasing temperature until it reaches the atmospheric pressure at the boiling point of the liquid.
With that background, we can now discuss how the vaporizer in an e-cigarette works. The amount of ambient vapor (in the surrounding air) of the nicotine solution in e-cigarettes is very low so turning it into vapor is simply a matter of letting it sit for long enough. The amount of time it would take is too long for a smoker to wait though so they use two tricks to speed up the process. 580
1) The fluid is heated so that the vapor pressure increases, and 2) a wick is used to drastically increase the surface area between which the air and fluid interact.
The heating of the fluid has the desirable side effect of making the cloud of vapor visible too. The best way to understand this is to think about a boiling pot of water. The water is converting to a gas at the surface and rising away from the pot, but you can't see water in its gaseous stage. You can, however, see steam because the water gas is mixing with the cool air above the pot and small droplets of water are condensing, much like a cloud. The same thing happens in an e-cigarette. Local heating in the vaporization area heats the air along with it and produces a vapor which you can't see. When this vapor mixes with the cool air as you inhale, it condenses to tiny droplets which give it the desirable appearance of smoke. So, when you blow out after puffing you are actually blowing out a little nicotine filled cloud.
As for the term atomizer, it does have a semi-scientific definition. An atomizer is something which creates a fine suspension of liquid droplets in a gas. This is exactly what I described to you above; the vaporizer turns the liquid to vapor which then becomes a suspension as the vapor saturated air cools. 580