If we sense the electromagnetic field at two points in space (with antennas) and multiply the two signals (with a computer), some of the time the product will be a positive number and sometimes it is negative. The product depends on the relative phase of the two signals, ie, whether - by the time they reach the computer - the two radio waves have their peaks and troughs aligned (in-phase) or if one wave has a trough when the other has a peak (out-of-phase). If you're confused, just draw some sine waves, either identical or with one flipped upside down, and then think about what happens if you multiply two numbers with the same sign together or two numbers with the opposite sign.
Apart from instrumental effects, the relative phase of the signals depends on the direction of arrival of the radio wave and the baseline distance between the antennas. If the wavefront hits the two antennas at the same time, or the delay is an integer multiple of the wave period, then the two signals will be in-phase. If the delay is a multiple of half the period the signals would be perfectly out-of-phase. If the delay is a multiple of one quarter wavelength then one signal will be zero at the same time that the other one is at its maximum value, and the product will be zero - the phases are in quadrature.
This means for any given baseline distance we can picture a pattern on the sky, where some positions will give a positive interferometer output and others will give a negative output, with a smooth transition between them (the top half of image below tries to illustrate this). In the case of an astronomical radio source, Earth rotation will move the source through the interferometer's beam, giving a quasi-sinusoidal interferometer output (shown in the bottom half of the image below). This pattern is called a fringe pattern or sometimes interferogram.
The first cool property of an interferometer is that you can use the fringe pattern to determine the astronomical position of the radio source, there is an example in the extragalactic observations page.
There are more "lobes" in the fringe pattern when the baseline is longer. It's not too hard to see why, picture an east-west baseline 10 wavelengths long. If a radio source rises directly in the east the signal will hit one antenna 10 wave periods before it hits the other. Later in the day, when the source is about to set in the west, it will be hitting opposite antenna 10 wave periods earlier.. so throughout the day the relative phase has to wrap around 20 times, so there must be 20 lobes in the fringe pattern. If the baseline was only 5 wavelengths long there would only be 10 phase wraps, which implies the fringes would be twice as slow.
The image below shows the same period of solar activity using two 20MHz interferometers with different baseline distances. The top panel shows the (non-interferometer) output of the four receivers that make up the two interferometers, they all look much the same. The bottom panel shows the interferometer responses, you can see that the 30m baseline (blue) has much slower fringes than the 135m baseline (red).
The images below shows a pretend radio source and the beams for a 20MHz interferometer with a 20m baseline (left) and a 150m baseline (right). As the source moves through the beam of the 20m baseline it will clearly occupy either a positive or negative lobe at any time, but not both, and we will record a classic fringe pattern. Whereas with the 150m baseline it is obvious that no matter where in the source is, it will occupy both positive and negative lobes at the same time, and we will not get a strong detection with the interferometer.
The second cool property of an interferometer is that you can measure the size of radio sources, by extending your baseline until you start to detect less power from the radio source. When it starts to become resolved then you know the source is the same size as your interferometer lobe spacings - which you can easily calculate.
The third cool property of an interferometer also exploits this property. By using long baselines we can resolve the the plane of the Milky Way so that it "vanishes" from the interferometer's view. We can then detect compact, discrete radio sources, such as extragalactic sources, that are normally obscured by the Galactic radio noise.
The computer uses a soundcard to sample the output of each receiver. The interferometer response is calculated by multiplying corresponding samples from the two receivers, and averaging this product for a second or so. This multiplication, or cross-correlation, process has a nice feature that the averaged response only includes the signals that are common to both receivers.
The fourth nice property of a (cross-correlating) interferometer is that signals found in only one receiver, such as the internal noise from within each receiver, will not be present in the (averaged) interferometer output, which enables more sensitive observations.
One of the most common questions I get is "why do you have two receivers and multiply the signals, can't you just connect spaced antennas and plug them into the same receiver?". Interferometers with two antennas that are combined like this are called adding interferometers. Adding interferometers are easy to make but lack most of the properties that make interferometry so useful.
You see, the interesting term for interferometry is the product between the antennas, V1*V2. After we square the output of an adding interferometer (to get a power measurement) the output is (V1+V2)^2 = (V1+V2)*(V1+V2) = V1^2 + V2^2 + 2*V1*V2. The V1*V2 term is present here, but it is generally weaker than the output from each antenna, the V1^2 and V2^2 terms.. which have nothing to do with interferometry. The output from this kind of interferometer usually looks like the output of a single antenna, with a weak set of fringes added to it. That said, people still use adding interferometers successfully, mostly at high frequencies where the diffuse Galactic emission is not as strong.
As an experiment for Joachim Koppen I once reconfigured Simple to work as an adding interferometer for a few days and obtained the data shown below. Interference fringes are present as the little wiggles but you can see the output is hardly comparable with the cross correlating interferometers. This data has a 82.5m baseline.