Further expanding upon the computer simulation idea brought up in MattClarke's answer: One interesting idea is the idea of mind uploading. If human consciousness happens to be Turing-compatible, then it
would be theoretically possible to convert a human mind into a piece of software.
Then, just like normal software, the host computer would be able to arbitrarily suspend and resume the simulation program. Modern operating systems can run hundreds of processes simultaneously because it has a scheduler that is constantly suspending and resuming processes, giving each process a fair share of computing time, i.e. timeslicing.
The running process has no awareness of whether it has been sliced or not; it just continues normally. So in that sense, no time has passed for it, yet time has passed in the outside world. However, the process might realize that it is being sliced by observing skips in the system clock, for example. So there was a change. But if you completely sandbox the process, it would not be able to perceive time passing any differently than normal, no matter how often and how long you suspend it.
There's a science fiction novel called Permutation City that explores lots of interesting implications of simulated minds:
Within the story, "Copies" – digital renderings of human brains with complete subjective consciousness, live within VR environments after a process of "scanning". Copies are the only objects within VR environments that are simulated in full detail, everything else being produced with varying levels of generalisation, lossy compression, and hashing at all times.
Much of the plot deals directly with the "lived" experience of Copies, most of whom are copies of wealthy billionaires suffering terminal illnesses or fatal accidents, who spend their existences in VR worlds of their creating, usually maintained by trust funds, which independently own and operate large computing resources for their sakes, separated physically and economically from most of the rest of the world's computing power, which is privatized as a fungible commodity. Although the wealthiest copies face no financial difficulties, they can still be threatened because copies lack political and legal rights (they are considered software), especially where the global economy is in recession. Hence they cannot afford to retreat into solipsism and ignore what is happening in the real world.
The story also explores the ideas of time slowdowns and suspension in the context of simulation vs. outside world:
At the opposite end from the wealthy Copies are those who can only afford to live in the virtual equivalent of "Slums", being bounced around the globe to the cheapest physical computing available at any given time in order to save money, while running at much slower speeds compared to the wealthy Copies. Their slowdown rate depends on how much computer power their meager assets can afford, as computer power is traded on a global exchange and goes to the highest bidder at any point in time. When they cannot afford to be "run" at all, they can be frozen as a "snapshot" until computer power is relatively affordable again. A Copy whose financial assets can only generate sufficient interest to run at a very slow rate is stuck in a rut because he/she/it becomes unemployable and is unable to generate new income, which may lead to a downward spiral.
The concept of solipsism is also examined prominently, with many less-wealthy Copies attending social functions called Slow Clubs, where socialising Copies agree to synchronise with the slowest person present. Many of these less-wealthy Copies become completely deracinated from their former lives and from world events, or else become Witnesses, who spend their time observing (at considerable time lapse) world events unfold, at the cost of any meaningful relationships with their fellow Copies. A subculture of lower/middle-class Copies, calling themselves Solipsist Nation after a philosophical work by their nominal founder, choose to completely repudiate the "real" world and any Copies still attached to it, reprogramming their models-of-brains and their VR environments in order to design themselves into their own personal vision of paradise, of whatever size and detail, disregarding slowdown in the process.