What are the types of acceleration?

To understand the types of acceleration, we must understand that acceleration is the rate of change of velocity with time.
The change in velocity can be either positive or negative, both situations referring to a type of acceleration ( later one sometimes being termed as deceleration or retardation).
Another point worth noting is that since velocity is a vector quantity (having both magnitude and speed) even though there is no change in speed, a change in direction would mean a change in velocity, and thus the object would be accelerating without any speed change. This is known as centripetal acceleration.
Another basis of categorisation can be the amount of change in velocity in equal intervals of time. If change is equal in equal intervals of time, we have uniform acceleration, if the change is different over equal intervals of time we have non uniform acceleration.
In non uniform acceleration, value of acceleration at a specific point in time (can be calculated by velocity vs time graph) is the instantaneous acceleration at that point and the average taken for whole range of time is average acceleration.
There are few other types of acceleration based on the driving force such as gravitational acceleration.
Further Reading
http://www.knowledgeuniverseonline.com/ntse/Physics/Types-of-Acceleration.php

Acceleration is defined as a rate of change of the velocity of a moving object:
veca = (Delta vecv)/(Delta t)
Note that acceleration, like velocity, is a vector quantity.
When an object moves along a curved trajectory (path), its velocity, in general, changes in both speed and direction. The velocity vector is always tangent to the trajectory. The acceleration vector can point anywhere, and its direction depends on whether the object is speeding up or slowing down.
To describe the acceleration vector when an object is at a given point on a trajectory, it is convenient to break it up into two components: one tangent to trajectory (and co-linear with velocity) at this point, and another one perpendicular to the tangent to trajectory, pointing toward the center of its curvature. The first component is called tangential acceleration, and the second one is centripetal acceleration.
The tangential acceleration indicates the change in the magnitude of the velocity vector, or speed. Thus, if the speed of the object is constant, the tangential acceleration is zero. Otherwise, it equals the rate of change of speed:
a_t = (Deltav)/(Deltat)
The tangential acceleration component points in the same direction as velocity if the speed is increasing, and in the opposite direction if the speed is decreasing. For circular motion, it can also be related to the angular acceleration alpha
through the formula
a_t = alpha*R , where R is the radius of curvature.
The centripetal acceleration indicates the change in the direction of the velocity vector. The greater the curvature of the trajectory, the more extreme the change of the direction of velocity. Thus, the centripetal acceleration is inversely proportional to the radius of the trajectory at the given point. It is also proportional to the square of the speed:
a_c = v^2/R
The centripetal acceleration always points toward the center of the curvature of the trajectory. An object moving along along a curved trajectory always has non-zero centripetal acceleration. The only motion with zero centripetal acceleration is motion along a straight line.
http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html