good answers, Nick and solrey.
Here is another way of looking at it. There are electric fields and there are magnetic fields. There are charged particles( with mass) like electrons and protons and ions, and neutral particles such as atoms, molecules and neutrons.
Well, neutrons don't last long outside the nuclei of atoms, but still, they are not charged or polarized. Because neutral particles aren't charged or polarized, they react to a gravity field, but not to a magnetic or an electric field. We are therefore talking charged particles when we are talking about electric currents. An electric current by definition is moving charged particles.
Let's posit that your lab is at the center of an arbitrary coordinate system, so if you have some charged particles moving through or past a given fixed position in your lab coordinate system you will have a current. If you can hold an electron say, for simplicity, in one place, or a 'cloud of electrons' in one area so that there is no net flow in or out of the contained area, you do not have a current. Random thermal movements—Brownian motion—do not constitute a current flow. Something has to get your quisecent, non-moving electron moving, or better, a bunch of them moving, to get a current started. Can a magnetic field do that?
Short answer, no. A magnetic field operates on an electron by changing its direction, not its speed. If an electron moves into a magnetic field, or moves within a magnetic field, the direction of its velocity vector (i.e.its direction and speed scalars) is changed but its speed vector remains constant to conserve kinetic energy. If the electron has no net velocity to begin with, a direction change cannot be imposed on a zero vector. If the electron is moving along and enters a magnetic field, the pointing direction of its velocity vector starts changing direction. If the entry path is at right angles to the orientation of the magnetic field, the electron (if the field is large enough) will just go into a circular loop whose radius of gyration is a product of its initial speed, its mass, and the strength of the magnetic field. If it is a charged thing with more than one basic electron charge, its total charge is used.
If the moving or drifting electron enters the magnetic field obliquely across the lines of magnetic force, instead of just looping in one plane normal to the lines, its vector components normal to the field and parallel to the field will cause its looping to move sideways, resulting in a helical path around an imaginary cylinder whose axis is parallel to the magnetic lies of force. This is a current, but remember, the electron had to enter the field with a scalar speed component first. It already "was a current" by moving and being charged.
So what can start an electron moving that is not a random collision process or a like-charged particle approaching it? An electric field.
If you put two plates in an electrical circuit so that there is a voltage potential between them, you have created a capacitor of one type. Other types include the dielectric cylindrical capacitors soldered into circuits and double layers or "sheaths" in cosmic plasmas. In this case, there is a surplus of charged particles in one location, and a deficit in another (the two plates, or two sides of a double layer. Between these two separated charges an electric field is set up. HOW it is set up is not too germane here - we just want to set up an experiment in this lab with a generic electric field.
Now take your thumb and forefinger and carefully move the electron into the electric field; stop, and let it go. What will it do? It will have a force exerted upon it in a straight line by the E field. It will accelerate toward the positive side of the separated charges. It will have that same force impressed upon it throughout its flight (until it bonks the positive plate and disappears into the circuit wiring) so that it is constantly accelerated. A large-mass charged particle acted upon by a weak electric field may have to be accelerated for some time before it achieves a significant fraction of the speed of light. A strong field, and a low-mass charged particle may become relativistic in microseconds or less.
So long as a charged particle is in an electric field, the field will try to add to its speed along the direction lines of the field. If the electron is in the company of other electrons which are also being accelerated, they will tend to fend each other off (like charges repel each other as they get close) so the electrons all become more and more aligned with the electric field lines, and keep going faster and faster. A field aligned current like this may be thermally very cool relative to neutral particles which may be bouncing around in there with them, because the neutrals are unaffected by the field. They, too, through actual collisions with the charged particles, can get themselves moving with the stream of charged particles, too.
In an old CRT TV the electrons are accelerated by an electric field and steered by controllable electromagnets to hit certain spots on the screen's phosphors, which give off various colors of light and show you Dancing With The Stars or something.
Electric fields in space can be extremely powerful, accelerating particles and creating fast moving protons that we call cosmic rays, good at creating micronucleation particles in the lower atmosphere upon which water vapor condenses and creates our clouds; and that controls long-term climate. —as a little aside there; I'm reading The Chilling Stars by Svensmark. Highly recommended.
Electric fields are what mainstream scientists use to accelerate their charged particles in their accelerators (hardly being dummies, they know not to try to accelerate neutral particles, but use them as the targets, instead), like the Large Hadron Collider when it is actually working. If the accelerator is straight like the linear accelerator at Stanford it doesn't take much in the way of magnetic fields to keep the particles moving where they want them. In a circular or racetrack accelerator it takes extremely powerful electromagnets to impart the required curve to the charged particles' trajectories and keep them inside the curving evacuated tube. If the electromagnets fail or get magnetically yanked off their supports (like at the LHC) it is "an engineering design error" and nobody looks good, and the electron beam plows into the walls of the curving tube and leave ugly, long electron scrape marks. Scree-e-e-eech!!!*#?##!!??????? !
So if you are wondering which kind of field gets electrons moving so as to create a current, it's an electric field somewhere, created by a charge differential. Electric fields are the engines of the plane, and make it go; magnetic fields are the elevators, ailerons and rudder. They can't do anything unless the plane is moving first.