What is the definition of a magnet or a magnetic field? Electric forces are the forces which come about between two types of charges, positive and negative. Gravitational forces are the forces between matter. Nuclear forces are the forces which act on the atomic scale and are quantum mechanical forces, they act between nucleons.
And, magnetic forces are the forces between magnets? I felt this definition wasn't specific enough so I searched up the definition of a magnet.
"A magnet is a material or object that produces a magnetic field" Well, what is a magnetic field?
"A magnetic field is a vector field that describes the magnetic influence on magnetic materials"
I feel like this is a loop.
I am sorry, but I am not satisfied and feel like there is a more fundamental definition of magnetic field or magnet that I am not aware of. So, what is a magnet?
 A: Electric and magnetic forces are tightly intertwined.
We can use the Lorentz force acting on a charge $q$
$$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$$
to define the electromagnetic field.
The force has two parts:

*

*The first part ($q\mathbf{E}$) of this force
is independent of the velocity of the charge.
We call it electric force, and actually this relation
serves as the definition of the electric field $\mathbf{E}$.

*The second part ($q\mathbf{v}\times\mathbf{B}$) of this force
is proportional to the velocity of the charge.
We call it magnetic force, and actually this relation
serves as the definition of the magneticic field $\mathbf{B}$.

It may be even more instructive to look at the force between two charges
$q_1$ and $q_2$
(moving with velocities $\mathbf{v}_1$ and $\mathbf{v}_2$,
separated by distance $\mathbf{r}$).
Neglecting any retardation effects by the finite speed of light,
this force is:
$$\begin{align}
\mathbf{F}&=\frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}\hat{\mathbf{r}} \\
  \\
 &+ \frac{\mu_0}{4\pi}\frac{q_1q_2}{r^2}\mathbf{v}_1\times(\mathbf{v}_2\times\hat{\mathbf{r}})
\end{align}$$
Here again the force has two parts:

*

*The first part of this force
is independent of the velocities of the two charges.
This is the well-known electric force as described by Coulomb's law.
Like charges repel, and unlike charges attract each other.

*The second part of this force
is proportional to the velocities
$\mathbf{v}_1$ and $\mathbf{v}_2$ of the two charges.
We can call it the magnetic force between the charges.
For parallel $q_1\mathbf{v}_1$ and $q_2\mathbf{v}_2$
the charges attract,
and for antiparallel $q_1\mathbf{v}_1$ and $q_2\mathbf{v}_2$
they repel each other.

Hence, the essence of the above is:
The electric forces come about between charges,
regardless of whether these charges are at rest or moving.
The magnetic forces come about between charges,
when both the charges are moving.
So instead of the unsatifsfying definitions

"A magnet is a material or object that produces a magnetic field"
"A magnetic field is a vector field that describes the magnetic influence on magnetic materials"

we can come up with better definitions like this:
A magnet is a material or object with lots of charges moving with similar
velocities or spinning around similar rotation axis'.
A magnetic field is a vector field that describes the velocity-dependent effect on moving charges.
A: 
Electric forces are the forces which come about between two types of charges, positive and negative.
Gravitational forces are the forces between matter.
Nuclear forces are the forces which act on the atomic scale and are quantum mechanical forces, they act between nucleons.

To finish your list, magnetic forces are the forces between aligned magnetic dipoles of subatomic particles.

"A magnet is a material or object that produces a magnetic field" Well, what is a magnetic field? "A magnetic field is a vector field that describes the magnetic influence on magnetic materials" I feel like this is a loop.

Magnetic fields are obtained in two ways:

*

*in a permanent magnet, a part of its particles (electrons, protons, neutrons) is aligned and holds itself up to the specific Curie temperature of this material

*accelerated (circularly moving or linearly accelerated) electrons also align themselves with their magnetic dipoles and realise a common magnetic field.

By the way, the magnetic moments of the electron and the other atomic particles are intrinsic (constants independent of external circumstances). And you get an insightful idea of magnetism and electromagnetic induction if you put all these phenomena in relation to the fact that electrons are not only a charge but also magnetic dipoles.
A: There is a very good explanation of how electric charges in motion create the thing we call a magnetic field in this paper, "Magnetism, Radiation, and Relativity", Supplementary notes for a calculus-based introductory physics course by Daniel V. Schroeder, Weber State University (http://physics.weber.edu/schroeder/
dschroeder@cc.weber.edu).
I struggled for years to understand this topic and I did not make any real progress until I read this paper. Schroeder's basic point is that moving charges exert special forces on charges that are not moving, and we call those special forces "magnetism" for convenience. I recommend you have a look at the paper and let us know if this helps.
A: Just a general comment on

"A magnet is a material or object that produces a magnetic field" Well, what is a magnetic field? "A magnetic field is a vector field that describes the magnetic influence on magnetic materials" I feel like this is a loop.

This way of thinking will cause you a lot of grief. Human insight, and especially physics, is full of cyclic statements like this one. The point is, that this is actually not a purely mathematical/logical statement. Logic is connected to experience by example. Children know very well how to learn by example, but as we grow older, we sometimes/to some extent lose that ability, and start being obsessed by words and logic.
It is actually pretty simple to resolve the cyclic nature of the definition. Take into your right hand one piece of magnetite someone found somewhere, say, in Bolivia. Take another piece of magnetite into your left hand and see/feel how it attracts/repels the first one and how orientation plays a role. Call the magnetite crystal a "permanent magnet". Take a piece of iron, see how it is attracted/repelled by the magnetite, but the orientation of the iron being irrelevant. Call the iron an "induced magnet". Identify approximate orientation marks of the permanent magnet as "magnet poles". Identify the force on one pole (with the other pole being sufficiently far away) with a hypothetical "magnetic field". And so on, and so forth...
