The only computing device presented in Inception is a dream sharing device that allows multiple people to enter a person's dream. The  layout of the person's dream can be manipulated; a person known as an architect is able to design the layout of the dream that the person's subconsious fills in. For example, the architect may add a safe into the layout, which the person's subconscious fills in with personal secrets. 

It is important to note the characteristics of the dreams that the dream-sharing device is able to share. Time slows within dreams that feel real; only when someone leaves the dream do they realize that they were in a dream. The dream-sharing device also allows for deception, dreams can be layered; the dream-sharing device can be used within a dream, allowing users to enter a dream within a dream. 

Inception provides no decsription of how the device actually works. The audience is simply shown the technology used. 

The idea of interacting with people in their dreams or influencing their dreams has been around for a long time, and while Inception goes far beyond what is actually possible, some technologies and early research exists moving towards such a reality. For example, one study was able to establish bidirectional communication between the dreamer and experimenter [1]. While we aren't quite able to simply hook up to a machine and enter each other's dreams, this is a step towards such a technology. However not every technology in this realm is a success, as is the case with something called Dormio [2]. Dormio was supposed to be a dream incubation device that would allow researchers to influence the dreams of the test subjects. While a very interesting concept, the study showed no real success, being closer to fantasy than an actual reality. Though it's success was limited, Dormio shows us the beginnings of what could one day be a technology such as the dream sharing device featured in Inception. All in all, while the dream sharing device portrayed in inception is unrealistic for current day technology, it very well could one day be possible.

It is undeniably a little bit juvenile to comment on how realistic a sci-fi fantasy film is, so this section will focus on the bigger picture instead of specific elements.

Importantly, from a realistic perspective, with technology like the mind reading and writing devices seen in Inception, an objectively superior approach is literal mind reading and writing instead of dealing with dreams. Such technologies have been attempted, such as the CIA's infamous MK Ultra, but no such technology has been perfected. With that being said, the human brain is still just an analog computer that can be decoded, as evidenced by the aforementioned use of MRI machines in various neurological studies regarding dreams. As such, the notion of brain hijacking for nefarious means isn't technically impossible, although, it is seemingly infeasible at the present, given the lack of documentation on the subject.

Another aspect worthy of analysis is entering dreams. While above sources mention communication with dreamers (in fact, such communication is likely already known of by most people anecdotally), the very sci-fi technology of "syncing brain waves" has very little grounding in reality, at least when modern technology is the subject.

Brain "writing" tech (as opposed to reading) is also simply not common in publications (presumably due to the dangers associated), so it may be less a question of infeasibility and more of a lack of documentation. If brain writing were something that could be done safely, the use of artificially generated hallucinations for military exercises or entertainment, as seen in the film, could certainly be physically possible, as all sensations are simply brain signals, however we can only speculate about such things.