Time dilation refers to the fact that clocks moving at close to the speed of light run slow. Consider two observers, each holding an identical clock. These clocks work using pulses of light. An emitter bounces light off a mirror, and the reflected pulse is picked up by a detector next to the emitter. Every time a pulse is detected, a new pulse is sent out. So, the clock measures time by counting the number of pulses received; the interval between pulses is the time it takes for a pulse to travel to the mirror and back.
If our two observers are stationary relative to each other, they measure the same time. If they are moving at constant velocity relative to each other, however, they measure different times. As an example, let's say one observer stays on the Earth, and the other goes off in a spaceship to a planet 9.5 light years away. If the spaceship travels at a speed of 0.95 c (95% of the speed of light), the observer on Earth measures a time of 10 years for the trip.
The person on the spaceship, however, measures a much shorter time for the trip. In fact, the time they measure is known as the proper time. The time interval being measured is the time between two events; first, when the spaceship leaves Earth, and second, when the spaceship arrives at the planet. The observer on the spaceship is present at both locations, so they measure the proper time. All observers moving relative to this observer measure a longer time, given by:
time dilation: Δt_{ }  =_{ } 

In this case we can use this equation to get the proper time, the time measured for the trip by the observer on the spaceship:
Δt_{o} = Δt  γ^{ }  = 10 (1  0.95^{2} )^{1/2} = 3.12 years. 
So, during the trip the observer on Earth ages 10 years. Anyone on the spaceship only ages 3.12 years.
It is very easy to get confused about who's measuring the proper time. Generally, it's the observer who's present at both the start and end who measures the proper time, and in this case that's the person on the spaceship.