1) Brownian motion - this is motion due to heat. Remember, electrons tend to be bounded to atoms, but as you increase the heat, the electrons may leave an atom and travel freely, but this motion will be random and though a single electrons move, the sum of all electrons motion is essentially zero. This motion is a source of electrical noise, a small random current that exists in most everything.
2) Motion in response to an electric field - applying an electric field will place a force on each electron in an object. This force may not be strong enough to induce any electron to leave an atom. It may simply cause an atom or molecule to polarize (i.e. The electron will tend to stay on one side of the nucleus). Or, it may cause the electron to move away from the nucleus in the direction induced by the electric field. In a good conductor, electrons are moved by even weak electric fields.
3) The free electron will likely encounter another atom or nucleus and need to be dislodged again. This process absorbs energy and can add to the Brownian motion in other atoms (i.e. create heat). This effect is generally called resistance.
4) In a crystal lattice, the atoms are arranged in a geometric pattern due to weak bonds of the atoms and molecules in the lattice. The lattice can carry charge as electrons or "holes" (absence of electrons) move from bond to bond between the atoms. This movement is again directed by an electric field.
5) It is beneficial to remember that individual electrons are subatomic particles that are traveling quite fast. As such they exhibit both particle and wave properties. It is impossible to know the speed and location of an individual particle. When we speak of current, we are speaking on the aggregate. So, one Ampere of electrical current means 6.23x10^24 electrons pass through at the point of measurement. We don't know each electrons speed or direction, but we know they passed through within the bounds of the object at the point of measurement.