What is reliability? Reliability is how consistently a test measures what it attempts to measure. Why is consistency important? Because when you measure something with an instrument two times, you want it to come out with the same answer (or close to it) both times. (This is called test-retest reliability, and it is an important measure of any kind of scientific testing.)
Personality is qualitative and therefore difficult to measure, so psychological instruments cannot have the same consistency you would expect from, say, a ruler. But there are generally accepted standards for psychological instruments. The MBTI® instrument meets and exceeds the standards for psychological instruments in terms of its reliability.
Validity is the degree to which an instrument measures what it intends to measure, and the degree to which the thing that the instrument measures has meaning.
Why is this important? If personality type is real (or rather, if it reflects the real world with accuracy), then we should be able to use MBTI type to understand and predict people's behavior to some degree. Type should help us differentiate the values, attitudes, and behaviors of different people.
Many studies over the years have proven the validity of the MBTI instrument in three categories: (1) the validity of the four separate preference scales; (2) the validity of the four preference pairs as dichotomies; and (3) the validity of whole types or particular combinations of preferences. Many of these studies are discussed in the MBTI® Manual (published by CPP).