'Nines' are
shorthand for describing the actual or (usually) required
reliability of a
system. Each 'nine' represents a decimal place in the
probability that the system will be available/online/functioning/whatever. So, for example, a system that is
four nines reliable (or four nines available) will be working with a probability of 0.9999, or 99.99% of the time.
Reliability nines are using in planning and architecting systems of all sorts, ranging from computer networks to spacecraft. They are a good tool for understanding how complex systems that perform a simple task can become so expensive in terms of resources; recall that a system's aggregate reliability is the product of all its subsystems' reliabilities. Therefore, if you have a system that is made up of four subsystems, and each subsystem is four nines reliable, the system as a whole is only 0.9992, or three nines reliable! In systems such as the Space Shuttle, where literally tens of thousands of subsystems are present, it is nearly a physical impossibility to approach reliability numbers that are acceptable without using massive redundancy and overengineering - both of which are extremely expensive both in resources and in system 'payload' performance.
Since each nine represents a geometric increase in the required performance, the cost of adding nines to the reliability of a system is usually roughly equivalent to a tenfold increase in cost per nine.