What happens in my retina if I press on my eyeballs? If I press my eyes I can "see" all kind of things: sparkling blue dots (which sometimes seem random and sometimes there seems to be a pattern in them), growing or diminishing rings of all kinds of color (I once read that these circles are also present in the retina of the non-yet-born, to provide some preparation), vague black-and-white faces, and many more, sometimes strange, sometimes for a short time recognizable forms. 

So, what strange capers are the different light receptors, or neurons in or behind the retinas of my eyes performing?
Short answer
Pressure phosphenes are believed to be induced by sensory neurons in the retina downstream from the photoreceptors due to stretch-mediated activation.

Background
You are referring to pressure phosphenes. They are visual perceptions induced by applying pressure to the eye ball. They are often described as a glow with arcuate or circular characteristics and are generally perceived in the visual field opposite to the area of pressure. If the object applying the pressure is small, the center of the perceived area appears light with a dark surround and a bright outer portion. However, there is substantial inter-individual variability in how pressure phosphenes are perceived (Chew et al., 2005). 

Eyeball deformation leads to an activation of ON-center ganglion cells in the retina, while OFF-center ganglion cells are inhibited. It is thought that the activation of ON-center and inhibition of OFF-center ganglion cells by eyeball deformation are caused by retinal stretching, which may also lead to horizontal cell stretch. Stretching the horizontal cell membrane probably generates an increase in membrane sodium conductivity and a depolarization of the membrane potential. This depolarization of the horizontal cell membrane potential is transmitted either directly or indirectly (via receptor synapses) from the horizontal to the bipolar cells. The bipolar cells, in turn, can activate or inhibit the ganglion cells (Grüsser et al., 1989). Note that in this model, the photoreceptor cells in the retina are not involved - it occurs downstream in the secondary sensory neurons.

References
- Chew et al., Eye (2005) 19: 683–5
- Grüsser et al., Physiol Bohemoslov (1989); 38(4): 289-30