Mission Critical means tasks must be completed in a very very secure way. For example, banking bussinesses. All those missions not only need robust programs but also need fault-tolerant hardware systems, saying, disk-array, redundant system device etc. Systems claim to serve "7 days, 24 hours" are all "Mission Critical".
As to Real-time System, you can see the meaning from its name literally--it needs to react to all requests instantly according to some preset method. Often, it is used to call computer control systems in machines, in streamlines and etc. Nonetheless, nowadays it's also evoled to name system which is "Timing Critical"--just like Zigzag said. An example is Quoting-Broker system.