Speed is rate of change in position, measured in distance over time, while acceleration is the change of speed in a particular direction over time. A car can move at a constant speed around a curve, but it is still accelerating into the curve because the direction of its motion is changing. Acceleration is actually change in velocity, which is speed in one direction, rather than speed by itself.
Despite common usage, it's possible for speed to change without any acceleration in a given direction. In other words, as long as the speed changes enough that the velocity remains the same, no acceleration occurs. Acceleration can be either positive or negative, with positive values indicating an increase in speed in a particular direction and negative values indicating a decrease in speed in that direction. A negative value of acceleration is equivalent to a positive value in the other direction.
Any change in speed, velocity or acceleration requires that force be applied to an object. This is because of inertia, which is a property of matter where any body tends to maintain a constant state of motion unless disturbed. This means that a still object doesn't spontaneously move, and a moving object doesn't spontaneously change direction or speed.