If you represent a sinusoidal waveform in the complex plane (i.e. x axis is the real axis; y-axis is the imaginary axis) you end up with a vector of magnitude 'r' which continuously sweeps out a circle of given radius (with r = Exp[jwt], in this case the radius is 1 - the coefficient before the exponential.

With increasing time (t), the real value of the complex exponential will generate a cosine wave while the imaginary values will trace out a sine wave.

In general the imaginary portion of the complex exponential is not a physical observable, that is to say it is nothing more than a mathematical simplification. For example in solving differential equations with sinusoidal driving functions it is much easier to find trial solutions using an exponential as the derivative of an exponent is just another exponent. So usually the imaginary part only exists on paper while solving a problem, and then is dropped, as the real portion of the solution is the only meaningful part. But, there are some cases where the imaginary part is useful (and actually represents a physical concept).

The example I can think of is in optical properties of dielectric materials (examples include glass, semiconductors - optical fibers are a good example). In this case a traveling electromagnetic wave (light!) is represented by a sinusoidal function as it passes through a material, and while it travels through the material the wave is attenuated, that is to say that some of the energy is absorbed by the material. In order to represent this mathematically one adds an imaginary component to the dielectric constant of the material (if you do not know this is a property of the material that describes the interaction of the light with the atoms in the material); this would be Exp[-jwt]: the negative sign results in a decreasing exponential function. To actually find the power lost in the material one must square the field of the electromagnetic wave (the function that describes the sinusoidal light wave) and with (Exp[-jwt])² we get a real decaying exponential that nicely represents the power lost to the material.

Well, that was a long one. Sorry to put you through that, but your question is really a complex matter! (Pun intended). If you need more examples I suggest looking in a good E&M book, there should be lots of examples with Exp[-jwt]. So good luck to you.