How is Real-Time Computing defined?
Like many things in life, there is no one definitive definition. For an algorithm to run in real-time, some people on the theoretical side say that this means it will take constant time per 'something.' Now you have to decide what a 'something' is but let me give a concrete example. Let's say that the input arrives one symbol at a time and you want to output the answer to a query as soon as a new symbol arrives. If calculating that output takes constant time per new symbol then you might say the algorithm runs in real-time.    An example of this is real-time exact string matching, which outputs whether a pattern matches the latest suffix of a text in constant time per new symbol. The text is assumed to arrive one symbol at a time.

However, an engineering answer will be less worried about "constant time" and more worried about it happening fast in practice and in particular fast enough that the result can be used by the time it is needed. So for example in robotics, if you want to play ping-pong it is useful for the robot to be able to work out where the ball is and move to hit it as the ball arrives, and not after the ball has passed. The asymptotic time complexity of the underlying algorithms will perhaps be of less interest there than just the observation that the code works out the location quickly enough.  To give another example, if you want to render video and can do it at 25 frames per second then it is reasonable to say that the rendering is happening in real-time.  

So basically you have two answers. One for the theoreticians/algorithmists and one that just says that you are doing the work as you need it on the fly.

EDIT: I should probably add that one extra feature one should require of even a constant time algorithm is that the time complexity is not amortised.  In this context, real-time == unamortised constant time.