The neural systems that control eye movements are interesting because they form a closed, well-defined system that is in many ways a microcosm of the whole brain, yet amenable to very precise, quantitative study. And since where we look determines what we see, understanding eye movements is an essential preliminary to understanding vision. If oculomotor people sometimes appear slightly schizophrenic, it is because they have to think of eye movements in two ways at once: as a motor system (whose output is the position of the eyes relative to the head) and as part of a sensory system (where what matters is where the eyes are pointing in space, which technically called gaze).
This functional distinction underlies the classification of the types of eye movement.
The commonest eye movements are those whose function is, oddly enough, not to move the gaze at all. They come about because gaze is the sum of head position and eye position (taking the body as stationary). Since we are free to move our head, but our visual system objects if the retinal image moves across the retina too much, we have to compensate for the head movement by moving the eyes an equal and opposite amount in the other direction. These gaze-holding movements are driven either by the balance organs of the inner ear (the vestibular system; hence the name vestibulo-ocular reflexes or VOR for these kinds of movement), or alternatively they can also be driven by the retinal image motion itself, in a feedback loop. These are called optokinetic responses or OKR.
The other main class of eye movement comes about because the very centre of our retina, the fovea, is specialised for high-quality, full-colour vision; to see an object really clearly, that is where its image has to be. This means that to examine different objects we must deliberately shift the gaze to bring the eye on to the target. Since the resultant motion disrupts vision, we have evolved to make these movements as fast, and therefore as short in duration, as they can possibly be; they are called saccades. In addition, since we have two eyes, they need to be co-ordinated so that images of an object fall on exactly the same parts of the two retinae. For distant objects, this means that the two eyes must always move equally: the eyes are 'yoked' or conjunct. But as an object moves closer, the eyes must unyoke themselves, and converge to line up with the target: these are called disjunct or vergence movements.
Finally, even when fixating a stationary object, the eyes are not still, but are making continual small movements. These micro- or fixation movements are composed of three components: slow drift, rapid, small-amplitude tremor, and micro-saccades that bring the gaze back when the drift has moved it too far from the target.
Of course, in real life there are many objects in the world around us that might be targets for saccades, so in addition to neural machinery in the brainstem to get the eye on to a designated target, we also need mechanisms higher up for choosing what target to look at in the first place. Some of this selection seems to be done in the frontal part of the cerebral cortex, where there are units whose firing seems to reflect the processes of choosing and discriminating between targets. Work on saccadic reaction times in humans seems to suggest that the brain runs a kind of race between signals representing different possible targets, with more probable targets starting nearer the finishing post than less probable ones. There is also a huge random element, rather like a gratuitous random handicap, so that reaction times are very variable even when the stimuli and conditions are absolutely constant. This may well represent a deliberate mechanism for making sure our behaviour is not too predictable by our predators (and you may like to think of it as the neural mechanism behind our illusion of 'free will' - see One-way Cartesianism.)
Roger Carpenter's home page