Exponentiation
When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the repeated multiplication, because the exponent is . Here, is the "base", is the "exponent", and is the "power" or, more specifically, "3 raised to the 5th power".
Exponential distribution
In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) The length of a process that can be thought of as a sequence of several independent tasks follows the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).
Reliability theory and reliability engineering also make extensive use of the exponential distribution. Because of the "memoryless" property of this distribution, it is well-suited to model the constant hazard rate portion of the bathtub curve used in reliability theory. It is also very convenient because it is so easy to add failure rates in a reliability model. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.
In physics, if you observe a gas at a fixed temperature and pressure in a uniform gravitational field, the heights of the various molecules also follow an approximate exponential distribution, known as the Barometric formula. This is a consequence of the entropy property mentioned below.