When a uniform conductor (resistor) of length L is connected to a battery (voltage V) and is given a potential difference across its ends, there is an electric (electrostatic) field E created through the conductor. Then E = V / L.
Due to this potential gradient and field intensity, the charged particles - free electrons (in the conductor) ie., from the conduction band in the atoms - free themselves from atoms and start travelling towards the positive terminal of the battery. Electrons have a charge equal to e. they have a mass = m.
The electrons accelerate with acceleration a = e E / m, due to the force = e E on them. Due to collisions of electrons with one another , they lose a lot of momentum and energy (to heat) in each collision. But again they accelerate. They attain a net uniform and steady speed along the length of the conductor from negative terminal towards the positive terminal of the battery. This steady speed is called drift speed.
Thus, the movement of particles which move slowly along the medium, under the influence of a field and force is called drift. The speed is called drift speed.