The rise of computers in the 20th century, and especially their exponentially increasing computational capacity and speed, has gotten many curious minds to speculate as to whether it is possible at some point to create computers that can think. Those who believe in things like the computational singularity, such as David Chalmers, think it's just a matter of time before we have to bow down to our new mechanized overlords.
Here is a Philosophy Bites interview with Chalmers on just such a question:
Others, like philosopher John Searle, however, think that, given everything we know about computation, it is impossible, even in principle, for computers ever to think, no matter their computational capacity. To prove this point, Searle came up with what has come to be regarded, by supporters and detractors alike, as a classic thought experiment: the Chinese room, which you get to learn about in the following 60 seconds.
For more on questions of mind, consciousness, personal identity, etc., visit the Brainspotting tag.