Real-time refers to hardware and software systems which are subject to constraints in time. Some kinds of software, such as many chess-playing programs, are allowed as much time as needed to perform a computation such as the next chess move. Web servers are an example of software that people work hard to optimize for performance, yet performance problems rarely incur any serious penalty.
Often the concepts of Hard Real-time and Soft Real-time are used to distinguish between things that must be done in a certain time period (Hard) and things that can get away with minor fluctuations in latency (within bounds) without ill effect.
Real-time systems are typically used where there is some interaction with a physical system wherein the luxury of unlimited computing time is not possible. An example would be the software that maintains and updates the flight plans for commercial airliners. It would not be possible to offer modern commercial air travel if these computations could not reliably be performed in real time.
The needs of real-time software are often addressed with the use of real-time operating systems, which provide a framework upon which to build real-time application software.