For a while now the fraction of condensed matter physicists that think about electronic transport measurements have been interested in noise as a means of learning more about the underlying physics in systems. I thought it would be useful to give a sense of why noise is important. First, what do we mean by noise? As you might imagine from the colloquial meaning of the term, electronic noise manifests itself as fluctuations as a function of time in either the current through a system (current noise) or the voltage difference across a system (voltage noise). These fluctuations are distributed about some mean value of current or voltage, so the smart way to characterize them is by taking the average of the square of the deviation from the mean (e.g., <(I - < I >)2>, where the angle brackets denote averaging over time, and I is the current.). You can imagine that these fluctuations are distributed over all sort of time scales - some might be fast and some might be slow. The natural thing to do is work in the frequency domain (Fourier transforming the fluctuations), and then you can worry about the power spectral density of the fluctuations. For current noise, this is usually written SI, which has units of Amps2/Hz. If you evaluate SI at a particular frequency, then that tells you the size of the mean square current fluctuations within a 1 Hz bandwidth about that frequency. There is an analogous quantity SV [V2/Hz] for voltage noise. If the power spectral density is constant over a broad range of frequencies (up to some eventual high frequency cutoff), the noise is said to be "white". If, instead, there is a systematic trend with a larger power spectral density at low frequencies, the noise is sometimes called "pink".
In any measurement, there might be several kinds of noise that one must worry about. For example, your measuring equipment might show that the apparent SI or SV has several sharp peaks at particular frequencies. This is narrow band noise, and might be extrinsic, resulting from unintentional pickup. The classic examples include 60 Hz (50 Hz in many places outside the US) and its multiples, due to power lines, ~ 30 kHz from fluorescent lights, 540-1700 kHz from AM radio, 85-108 MHz from FM radio, etc. Extrinsic noise is, in physicist parlance, uninteresting, though it may be a major practical annoyance. There are sometimes intrinsic sources of narrow band noise, however, that can be very interesting indeed, since they indicate something going on inside the sample/system in question that has a very particular time scale.
There are three specific types of noise that are often of physical interest, particularly in nanostructures: thermal (Johnson-Nyquist) noise, shot ("partition") noise, and 1/f ("flicker") noise. I'll write a bit about each of these soon.