Phobia of clowns, or 'coulrophobia' presents as an intense irrational fear / anxiety of anything associated with clowns, or in some cases only certain types of clowns. It is referred to as a specific phobia, specific phobias are characterised by the presence of a very specific stimulus as the trigger, and in this particular case the stimulus happens to be a clown or clowns. No amount of reasoned thought seems to dispel the fear.
The phobia can generalise or begin to overlap into other areas, whereby the sufferer may then become afraid of going to a circus, or even seeing a circus on television. They may panic when they see someone wearing a mask that looks like a clown's face, or even if they see or smell greasepaint, or even when they see a photograph of a clown.
When the fear is at its strongest 'imprinting' takes place (imprinting refers to a very strong negative perception of the frightening experience), this is where the individual experiencing the intense anxiety learns very quickly to feel frightened or even terrified when exposed to whatever the phobic stimulus, which in this case happens to be clowns. Then anticipatory anxiety can develop even when thinking about clowns. Avoidance of all 'clown' situations / scenarios reinforces the phobia even more.
Phobias usually develop through having gone through an unsettling or even frightening experience, which generally takes place in childhood (not in every case though). Typically the experience is then repressed (taken from consciousness to a deeper part of the mind), where it becomes the source of the intense irrational fear. All phobias have one common denominator, which is they trigger the 'acute stress response,' sometimes called the 'fight or flight' response.