Summary: Neuroimaging scans and EEG readings comparing nonconscious and conscious stimuli are compared, showing particular patterns in conscious processes. These findings are in line with predictions made by the Global Workspace Theory of consciousness, in which consciousness is closely related to interaction between specialized modules of the brain.
When a bunch of photons hit your eye, it unleashes a long chain of cause and effect that leads to an image being mapped in your brain. When does that image become conscious?
Merikle et al performed experiments in the 80s which helped to resolve this question. In the Stroop task, people are asked to read words written in a different color than the word. Words written in their color (green) are easier to read than those not in their color (also red). Merikle modified the stroop task, using only two colors (red and green), and using the word to prime subjects to describe the color. As was expected, when "green" comes before a green square, subjects respond faster than with no priming.
However, when the situation is regularly reversed and the "red" prime normally comes before a green square (and vice versa) people also respond faster to similar levels. That is to say, subjects are able to notice that the prime and stimulus are incongruent, and act on that information to respond faster to the stimuli.
When the reversed prime ("red" before green) is flashed for such a short time span that people don't report seeing it, they are unable to use this information to react faster to the green stimulus, and the typical Stroop effect is observed -- being subliminally primed with a congruent color speeds up recognition, being subliminally primed with an incongruent color slows it.

- Mask a stimulus, by presenting it close in time to other unrelated or interfering stimuli. (i.e. a word flashed for 33 ms is noticeable by itself, but not when proceeded and followed by geometric shapes)2,3
- Use dichoptic masking, where you present two different images to each eye, and the subject reports seeing something which is neither of those4
- Use flash suppression, where you show one eye an image and flash shapes in the other eye to interfere with image perception5
- Use inattentional blindness, where you present something that participants aren't focusing on.
- Distract them! Present another stimulus and then quickly follow it with the one that you're interested in presenting preconsciously during their attentional blink.6

The idea that conscious access is related to recurrent processing in the frontoparietal region stands up to experimental verification. Researchers are able to interfere with conscious reports of information independently of stimulus identification simply by applying transcranial magnetic stimulation to the prefrontal cortex, without changing the stimulus.8
Notes
A huge thanks to John Salvatier for getting me a bunch of the papers and editing feedback and putting up with my previous attempts to write an article like this. Also thanks to mtaran, falenas108, and RS (you don't know him) for reading drafts of this article.
Images are from Zeki 2003 and Dehaene 2011, respectively. I'd be very happy if someone helped me format that to show up with the pictures.
1Merikle & Joordens, 1997
2Dehaene, S., & Changeux, J.-P. 2011
3Breitmeyer & Ogmen, 2007
4Moutoussis & Zeki 2002, Image from Zeki 2003
5Tsuchiya & Koch
6Marti et al 2010
7Lamme 2006
8Rounis et al 2010
9Baars 1997
10Metzinger
References
Baars, B. (1997). In the Theatre of Consciousness: The Workplace of the Mind. New York: Oxford University Press. Retrieved from here
Bruno G. Breitmeyer and Haluk Ogmen (2007) Visual masking. Scholarpedia, 2(7):3330
Dehaene, S., & Changeux, J.-P. (2011). Experimental and theoretical approaches to conscious processing. Neuron, 70(2), 200-27. Elsevier Inc. doi:10.1016/j.neuron.2011.03.018
Kouider, S., & Dehaene, S. (2007). Levels of processing during non-conscious perception: a critical review of visual masking. Philosophical transactions of the Royal Society of London. Series B, Biological sciences, 362(1481), 857-75. doi:10.1098/rstb.2007.2093
Lamme, V. A. F. (2006). Towards a true neural stance on consciousness. Trends in Cognitive Sciences, 10(11). doi:10.1016/j.tics.2006.09.001
Merikle, P. M., & Joordens, S. (1997). Parallels between perception without attention and perception without awareness.Consciousness and cognition, 6(2-3), 219-36. doi:10.1006/ccog.1997.0310
Lamme, V. A. F. (2006). Towards a true neural stance on consciousness. Trends in Cognitive Sciences, 10(11). doi:10.1016/j.tics.2006.09.001
Lau, H., & Rosenthal, D. (2011). Empirical support for higher-order theories of conscious awareness. Trends in cognitive sciences, 15(8), 365-373. doi:10.1016/j.tics.2011.05.009
Marti, S., Sackur, J., Sigman, M., & Dehaene, S. (2010). Mapping introspection’s blind spot: reconstruction of dual-task phenomenology using quantified introspection. Cognition, 115(2), 303-13. Elsevier B.V. doi:10.1016/j.cognition.2010.01.003
Metzinger, T. (2003). Being No One. Philosophy, 699. MIT Press.
Moutoussis, K., & Zeki, S. (2002). The relationship between cortical activation and perception investigated with invisible stimuli. Proceedings of the National Academy of Sciences, 99(14), 9527. National Acad Sciences. doi:10.1073/pnas.PNAS
Rounis, E., Maniscalco, B., Rothwell, J., Passingham, R., & Lau, H. (2010). Theta-burst transcranial magnetic stimulation to the prefrontal cortex impairs metacognitive visual awareness. Cognitive Neuroscience, 1(3), 165-175. doi:10.1080/17588921003632529
Tsuchiya, N., & Koch, C. (2005). Continuous flash suppression reduces negative afterimages. Nature neuroscience, 8(8), 1096-101. doi:10.1038/nn1500
Zeki, S. (2003). The disunity of consciousness. Trends in Cognitive Sciences, 7(5), 214-218. doi:10.1016/S1364-6613(03)00081-0
Nice article!
Folks who are interested in this kind of thing might also be interested to see the Koch Lab's online demos of CFS, which you can experience for yourself if you happen to have some old-style blue-red 3D glasses kicking around. This is the method where you show an image to the nondominant eye, and a crazy high-contrast flashing stimulus to the dominant eye, and the subject remains totally unaware of the image for (up to) minutes. Pretty fun stuff :) http://www.klab.caltech.edu/~naotsu/CFS_color_demo.html
You might also be interested in Giulio Tononi's "Integrated Information" theory of consciousness. The gist is that a brain is "conscious" of features in the world to the extent that it is properly causally entangled with those features, and has represents a large amount of information about the world in a deeply entangled way. Not easy to explain in a few sentences, but it seems to me to be a deeper theory that is perhaps related to this "Global Workspace" idea. I think you can find his most well-known paper at: http://www.sciencemag.org/content/282/5395/1846.short, many more available by poking around google scholar.
I can see both at the same time, and switch between different modes at will. I also tend to be immune to or see much more clearly all kinds of optical illusions and visual effects like this at various at will, or see them all the different possible ways simultaneously, able to see intuitively why things like this works, separately see "colours" like "movement speed" and "local contrast" in a way that intuitively feels like they were sent from the eyes as separate from maps of those properties that have more a feel of concious ... (read more)