I am again restating what has been said.
The amortized analysis formally introduced by Robert Tarjan involves considering the whole sequence of operation throughout the algorithm. The analysis is based on a simple practical phenomenon that when we run any algorithm there are some operations which are cheap and some are costly but overall the costly operations do not force the algorithm to perform very badly. There is simple charging argument involved that there are enough cheap operations that account for the extra cost paid for the costly operations.
An easy example includes when you are simulating a queue using two stacks.
A little involved analysis comes in disjoint set union problem using path compression which has inverse ackermann funtion complexity on an average.
It is important to note that it is not same as average case complexity analysis or the probabilistic analysis. In this we consider the worst case analysis of the algorithm as a whole (instead of each operation separately). Now since we are talking of worst case complexity, the analysis becomes important to prove the algorithm efficiency as the algorithm's performance will be upper bounded by the amortized complexity. However if we do take each operation's worst case analysis separately then our analysis might be quite loose as a whole.