The stereo base refers to the distance separating the two lenses on a stereo camera. It is measured from the center of the lenses. (If you are using a single-lensed camera and a slider bar, it refers to the distance separating the two lens positions.) For most stereo cameras, this is based on the average distance between a person’s pupils, usually in the 62-70mm range. This type of stereo imaging is known as ortho stereo, and is designed to mimic the view one would have if they were standing in the same spot that the photo was taken from. Ortho stereo is the kind of stereo image that our brains accept most naturally as reality, and in my view, the most interesting for this reason. However, there are reasons to sometimes work with different stereo bases.
One is close-ups and macro photography. When we are looking at something that is quite close to us, our eyelines converge inwards to keep the subject in clear view. As stereo camera lenses are fixed in a straight forward position and can’t do this, subjects that are too close to the camera look strange and jarring, the way your hand does when you hold it in front of your face while keeping your eyes on the horizon. One way we can get around this problem without pointing the lenses inwards (which can present its own set of optical issues) is to move the lenses closer together. Cameras with a slightly narrower stereo base are more suited for shooting close-ups, and cameras with an extremely narrow stereo base, such as the Macro Realist, are required for shooting macro images. Stereo photography with a stereo base narrower than that of ortho stereo is broadly referred to as macro stereo.
Another reason to use a different stereo base would be to exaggerate the dimensionality of distant subjects. Close objects always look more three-dimensional to us than faraway ones, as each eye is getting a more different view of (or seeing further around each side of) an object that is close. Mountains on the horizon basically look flat to us, as each eye is getting a near identical view. This is why when you shoot a stereo photo in which everything in the frame is distant, it looks indistinguishable from a regular, non-stereo photo. However, if you expand the stereo base, to a foot, a few feet, a yard, etc., you can get each lens to “see further around” faraway subjects, and to capture them in a way that feels more three-dimensional. This wide stereo base photography is known as hyper stereo. My understanding is that it can make these faraway objects feel smaller than they are, and that you can’t capture close subjects in the same frame. It’s usually accomplished with one or two single-lensed cameras on a long slider bar, and is most often employed for landscape photography.