It is not quite accurate to say you can measure acceleration. It is more accurate to say we measure things like time and distance and then use these measurements in order to calculate the acceleration of an object.
Consider a simple example of a trolley rolling down a slope.
You need to place a card on top of the trolley that is shaped like the graphic below:
We then place a light gate at the bottom of the slope and release the trolley.
The light gate is connected to a device that measures the time the light is blocked when the first section of the card passes through it. We input the length of the first section and the device calculates the initial velocity (u).
It repeats the measurement for the time the second part of the card blocks the light and calculates the final velocity of the trolley as it passes through the gate (v).
The device also measures the time between the first and second card to pass through the gate. This gives us the time between the velocities (t).
We now use the following equation to calculate the acceleration: