This is one of those pieces of industry specific jargon that has been slowly working its way into popular parlance for some time now. Its birthplace is the military-industrial complex, specifically the aerospace industry.
During the Apollo Project, critical path project analysis allowed project managers to identify the critical stages of a project's development. Similar systems analysis allowed engineers and project managers to identify critical points of failure with in a system or process, be it a communication satellite or an automotive assembly line. These points of failure, single elements of the overall systems, are referred to as "mission critical" if their failure means that mission success is no longer possible.
EXAMPLE #1: Your mission is to use your car to get groceries from the supermarket. If the battery has been stolen out of your car, you will be unable to start the car and get to the store. The battery is mission critical.
EXAMPLE #2: Same mission as above, except that you have a brand-new battery in the car. It cranks without a hitch and you're off. It's hotter than hell, so you turn on the air conditioning. Unfortunately, all the coolant has leaked out of your AC. While this makes you deeply uncomfortable in your vinyl pants and fringed buckskin shirt, you are still able purchase groceries and return home. Thus, the air conditioning is NOT mission critical
The term is used a great deal in the software industry. Typically, if a critical function of an organization is served by a software system, like a transaction server for an electronic retailer, then that system is mission critical. No transaction server = no sales = no cash influx = mission critical. Thus, systems analysts are paid a great deal of money by companies to locate these potential points of failure and figure out how to prevent them from failing. This "hardening" of the system can be accomplished in a couple of ways. First, you can create redundant systems] that are capable of carrying out the mission critical task, and build in failover (Failover is where the backup system or parallel system automatically picks up the slack from the primary systems failure). Second, you can "bulletproof" a single-point-of-failure system so that it just doesn't fail. This is one of the reasons that Linux and Unix are popular. When folks talk about software being "Enterprise Grade", its suitability for inclusion in a mission critical computing environment is one of the attributes they're talking about. Microsoft claimed that Windows NT 4.0 was a mission critical grade OS if administered correctly, but this is like saying a frog wouldn't bump its ass hopping if it had wings.