Uniform acceleration refers to constant acceleration. Since acceleration is nothing but rate of change of velocity, motion with uniform acceleration refers to a constant change of velocity. In other words,
a = dv/dt
and da/dt = 0 (i.e., rate of change of acceleration is zero, in case of uniform or constant acceleration)
where, a is acceleration and v is velocity.
One of the most common examples of uniform acceleration is the downward motion of objects on Earth. For example, when we drop something from a great height, it falls down due to Earth's gravity. The only force acting on the object is Earth's gravity and the acceleration is due to this gravitational pull and is equal to acceleration due to gravity or g (equal to about 9.8 m/s^2). The falling object will continue to gain velocity due to its acceleration, till it reaches terminal velocity. There can be other cases of uniform acceleration, esp. while driving a car, etc.
Hope this helps.