Can an electric field be converted into a magnetic field and how? If not, then why? My teacher says that a rest charge creates an electric field and a moving charge creates both electric and magnetic fields.
But, the cause of a magnetic field is velocity, so, is there something which changes an electric field into a magnetic field?
Question: Can an electric field be converted into a magnetic field? If yes, then how? If not, then why?
 A: The Ampere-Maxwell equation (also here):
$$\nabla × \vec{B} = \mu_{0}\vec{J} + \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t}$$
A current density $\vec{J}$ can cause a magnetic field.
A changing electric field can also create a magnetic field.
The latter can be reasoned as a consequence of charge conservation.
A: It depends a bit on what you mean by "converting". If your point is

Given a reference frame, can one describe the same Physics described using an electric field by using a magnetic field?

then the answer is not.
If your point corresponds to the interpretation given in jensen paull's answer, which is similar to

Given a reference frame, can an electric field induce a magnetic field or vice-versa?

the answer is yes, as mentioned in jensen's answer.
There is, however, other interpretations, one of which I find particularly suitable for your question:

Given that a phenomenon is described in terms of electric fields in one reference frame, could it be described in terms of electric and magnetic fields in other reference frames?

The answer is yes. The separation between what is an electric and what is a magnetic field depends on reference frame. If you consider the problem of the electromagnetic field of a charge, it is purely electric in the charge's rest frame, but a magnetic field appears in every other reference frame. Hence, you can "convert" an electric field into a magnetic field and vice-versa by changing reference frames. There is nothing paradoxical about that, for the forces experienced by charges are still the same. The only difference is that some observers will say the force is due to an electric field, while others will say it is due to a combination of electric and magnetic fields.
The restrictions on this sort of "convertion" is given by the Lorentz invariants of classical electromagnetism. More specifically, the quantities
$$B^2 - E^2$$
and
$$\mathbf{B} \cdot \mathbf{E}$$
are the same in all reference frames (I'm using Heaviside–Lorentz units with $c=1$). This restricts what you can make vanish on each reference frame. For example, if $\mathbf{B} \cdot \mathbf{E} \neq 0$, then there are no reference frames in which you have only an electric (or only a magnetic) field, for if there was you'd have $\mathbf{B} \cdot \mathbf{E} = 0$. Similarly, if you do have $\mathbf{B} \cdot \mathbf{E} = 0$, but $B^2 - E^2 > 0$, then there are no frames with only an electric field, for that would lead to $B^2 - E^2 < 0$, and so on.
In short, changes of reference frames can lead to a "mixing" or magnetic and electric fields. There are restrictions on how these conversions might happen due to the existence of combinations of magnetic and electric fields that have the same value in all inertial reference frames. For more details on how to compute these transformations, see, e.g., this Wikipedia article. Most E&M books also have a chapter in Special Relativity, in which these themes are often discussed.
A: The electric and magnetic fields are really just components of a single common structure that exists in 4-dimensional spacetime. The magnetic field is those components perpendicular to the time axis, and the electric field is those components running parallel to the time axis.
The figure below tries to show what the electromagnetic field looks like in spacetime. Time runs up the page, and the XY coordinates of space are horizontal. The Z coordinate is not shown.
If the particle is stationary, then it stays at the same horizontal (i.e. spatial) position as time passes, so the velocity looks like an arrow pointing straight up. The electromagnetic field is a set of planes radiating out from the charged particle's trajectory. The planes are entirely vertical, parallel to the time axis, so this is a pure electric field.
If the particle is moving, then it changes horizontal XY position as time passes, so the velocity arrow is tilted. The electromagnetic field is tilted along with it, so there is a horizontal component as well as a vertical component to each tilted plane. The horizontal component is the magnetic field B, and the vertical component is the electric field E.
The three components of E are represented by the timelike coordinate planes xt, yt, zt, and the three components of B are represented by the spacelike coordinate planes yz, xy, and xz. We normally drop the t from the E field ($F_{xt}\rightarrow E_x$, $F_{yt}\rightarrow E_y$, $F_{zt}\rightarrow E_z$) and take the perpendicular vector to the plane for the B field ($F_{yz}\rightarrow B_x$, $F_{xz}\rightarrow B_y$, $F_{xy}\rightarrow B_z$) to turn them into vectors.
You can get the same picture whether you are stationary and the charged particle is moving past you, or if you are moving past a stationary charged particle (tilting your coordinate system's time axis). They're the same thing looked at from different points of view.

