Acceleration and speed are not measured in the same units, because they are not the same physical quantities.
Speed is, roughly, the rate at which distance changes: `v = d/t` . As such, it has to be measured in units of length, such as meters, divided by the units of time, such as seconds.
Acceleration, however, is the rate at which the speed changes. Acceleration indicates how much the speed changed in a unit of time: `a = (Delta v)/(Delta t)` . In other words, it indicates by how many meters per second (or other units of speed) the speed changed in 1 second (or other unit of time.) So, the acceleration has to be measured by meters per second divided by seconds: `(m/s)/s = m/s^2` .
Hope this helps.