What are current neuronal explanations and models of 'consciousness'? I would like to understand more about consciousness from a neuroscientific perspective. I have a limited understanding of it in the philosophical/psychological sense through lectures.

Although it is hard to define, here is a definition from Christof Koch's website I will provide:


  "At this point in the scientific exploration of this phenomena, it cannot be defined rigorously. Consciousness usually (but not always) involves some form of attentional selection and a rapidly decaying form of information storage. For strategic reasons, most of the empirical research has focused on the brain states underlying conscious sensory perception, the neuronal correlates of consciousness, or NCC. I avoid taking any particular ideological position in the debate concerning the exact relationship between the NCC and conscious experience." 


I do not know what the debate is about but I'm curious about this relationship between NCC and overall conscious experience.

Rather than reading through every model of consciousness proposed, I was hoping someone who might be more knowledgeable to briefly explain or provide references to literature/reviews of models with a strong neuronal emphasis. In particular I would prefer models more so if they are not specific to a single mode of sensory perception.
The major neural models of consciousness at the moment roughly fall into two camps: cognitive and phenomenological. They are defined by controversy surrounding what types of experience qualify as concious.
Cognitive models
On the one hand there are strong cognitive models of consciousness, such as the one proposed by Stanislas Dehaene, where consciousness is characterised - neurally - by large scale, reverberant processing across the whole brain. That is, when feedforward stimulus based activity and top down feedback activity (i.e. internal, cognitive factors) are coordinated across the whole cortex. On this view, a stimulus can be said to be consciously perceived when it gains access to a special population of workspace neurons that have limited capacity and broadcast information related to the stimulus to other modular subsystems (e.g., memory, language sensory modalities). Limited capacity and broadcasting are properties of workspace neurons that account for both the limited nature of consciousness (e.g., attention), and coherency of consciousness (that is we perceive multi-sensory information coherently as events that are bound together, not as separate bits of visual, auditory and tactile experiences). It is believed that these workspace neurons are apart of the fronto-parietal attention network, and are involved in selecting information in the occipital-temporal systems.
Phenomenological models
In contrast, so called phenomenological theories, such as the those put forward by Victor Lamme and Ned Block, propose that consciousness also arises from local recurrent activity between two brain regions. The central feature of these theories is that - in addition to the cognitive forms of consciousness - there is also phenomenal consciousness which accounts for raw sensory (non-cognitive) properties of consciousness.
The key argument for this type of consciousness is that phenomenal consciousness overflows cognitive consciousness, that is, we are conscious of things that are outside of the focus of attention and that do not become encoded in a more durable form by the central cognitive system (I think Ned Block would still argue phenomenal consciousness requires attention so I am setting up a caricature here). According to this theory, phenomenal consciousness is supported by local recurrent activity between two regions. For example, it is possible to have a phenomenal sense of motion when motion selective cortex (area MT) and primary visual cortex (V1) enter into a feedforward and feedback relationship. Note Block and Lamme hold the same views as Dehaene regarding cognitive consciousness.
Old and other models
It is also possible to distinguish these two theories of consciousness from other dated views that argue that particular parts of the brain reflect the goings on of consciousness. For example, one common view is that the processing in the frontal lobes reflects consciousness. These regionally specific theories are quite different from the ones described above where consciousness is characterised by relationships between regions. However, these types of theories are very much out of fashion and for good reason - consciousness is likely to be a complex dynamic property of the system.
However, once you start to say that the whole system is important some people, such as Alva Noe, argue that it is pointless even saying that consciousness resides in the brain!!!! For Noe, consciousness is a dynamic relationship between the organism and the environment. I would not say that Noe's views are widely held in the neuroscientific community.
Well, these are the only models that I have followed in any detail. I know there are other more complex models, such as Giulio Tononi's Information Integration model, but this model requires a good knowledge of information theory (so I never followed it up). If any of this material needs more clarification I am more than happy to expand on it.

Bibliography
For an intro, try reading:

Kouider, S. (2009). Neurobiological Theories of Consciousness. In Banks, W. (Ed.) Encyclopedia of Consciousness. Elsevier, vol. 2, 87-100. (pdf)

See also:

Dehaene S, Changeux JP, Naccache L, Sackur J, and Sergent C (2006) Conscious, preconscious, and subliminal processing: A testable taxonomy. Trends in Cognitive Sciences 10: 204–211. (pdf)

Lamme VA (2006) Towards a true neural stance on consciousness. Trends in Cognitive Sciences, 10: 494–501. (pdf)