Does Lorentz force only refer to the force described by the Lorentz Law? I'm sorry for my very stupid question. I'd really appreciate some help. 
I'm just rather confused on what the Lorentz force actually is. So my question is, simply put - what is the Lorentz force? Does it only refer to the force described by the Lorentz Law (F=qvB)? 
Is it only applicable for a single particle where the force is perpendicular to the other involved factors? 
When two magnets repel each other, is the force that occurs that causes them to do so the Lorentz force? If not, what kind of force is it and what equation should be used to describe it?
 A: 
So my question is, simply put - what is the Lorentz force? Does it only refer to the force described by the Lorentz Law (F=qvB)? Is it only applicable for a single particle where the force is perpendicular to the other involved factors? 

The term "Lorentz force" is used in several different meanings. Yeah, I know.
Before Lorentz, correct formulae to calculate force on macroscopic charged or current carrying bodies were known (Ampere, Biot, Savart, Laplace knew how to calculate forces). What Lorentz did is to formulate a microscopic theory of charged particles in which matter is made of such particles. In this theory, force on a charged particle due to EM interaction was expressed as a function of:
1) density of charge $\rho$ and density of electric current flux $\mathbf j$ (usually called current density) and
2) quantities describing state of EM field $\mathbf d$ and $\mathbf h$.
The force was postulated to be the integral
$$
\mathbf F = \int_V \rho \mathbf d + \mathbf j \times \mathbf h\, dV
$$
where the integration is done over space region that contains all parts of the particle.
Later in 20th century, this notation fell in disfavour and $\mathbf d,\mathbf h$ were replaced by symbols $\mathbf e,\mathbf b$ but the meaning remained the same. This is one meaning of "Lorentz force".
This theory of Lorentz was quite useful and lead to successful models of interaction between matter and EM field (model of refraction, absorption, Zeeman effect and others). But this "Lorentz force" had a substantial problem: it could not explain how tiny charged particle of non-zero size could hold together. And even if some non-EM forces were added to hold it together, any relativistically correct model of such finite-size particle has infinity of degrees of freedom and is very hard to work with.
So, people very soon simplified the theory by assuming the charged particles are simply points. Then, the question of how the charge can hold together vanishes - there are no repelling parts. In this theory, the formula for force is postulated to be
$$
\mathbf F = q\mathbf e + q\mathbf v\times\mathbf b
$$
where $q$ is charge of the particle and $\mathbf e, \mathbf b$ is external electric and magnetic field acting on this particle.
Observations of electrons in EM field (J.J. Thomson's experiments and many later ones) confirm this expression describes force acting on electrons in external EM field very well. This is another meaning of the term "Lorentz force".
Sometimes, people deal with particles moving in a region of space where electric field is negligible and the only thing that matter is the magnetic force. Then, the formula simplifies into
$$
\mathbf F = q\mathbf v \times \mathbf b.
$$
This is also called "Lorentz force".
However, and unfortunately, this is not all.
Going back to macroscopic phenomena, force of magnitude $BIL$ acting on a wire of length $L$ carrying current $I$ in magnetic field $B$ is also called by some textbooks and people "Lorentz force". This usage seems quite common, but in my opinion, is incorrect*;  a better name would be the original terminology used before Lorentz: "magnetic force" or "ponderomotive force", which means roughly "force acting on/moving heavy matter", as opposed to "electromotive force" which roughly means "force moving electricity in a conductor".

When two magnets repel each other, is the force that occurs that causes them to do so the Lorentz force? If not, what kind of force is it and what equation should be used to describe it?

In macroscopic theory, no, because there are no charged particles. The force should properly be called "macroscopic magnetic force" and can be calculated either as:
1) force of one magnetic dipole on another magnetic dipole (the Coulomb approach)
2) force of one body carrying current loop on another body carrying current loop (the Ampere approach)
In microscopic elaboration of the second approach, the force on any body with a current loop is sum of the Lorentz forces $q\mathbf e + q\mathbf v\times \mathbf b$ on all particles that are part of that body.
* This force and formulae for it were known before Lorentz did his work. Also this expression is from macroscopic theory where the force acts on the heavy part of the body, not merely on charged particles.
A: The Lorentz force is the electromagnetic force on a charged particle, that is the resultant of the force on the particle due to (external) electric and magnetic fields. For a particle of charge q it is given by $$\vec F = q \vec E + q \vec v \times \vec B.$$ The second term is the Motor Effect force, that is the force due to the magnetic field. [The 'vector product' notation tells you that the force is in the direction given by Fleming's left hand rule.]
The magnetic force on a current-carrying circuit is the vector sum of the magnetic Lorentz forces on the moving charge carriers inside it. In this case it's more convenient to work in terms of current rather than velocity of charge carriers [For electrons flowing through a wire $I=-n A v e$ in which $n$ is the number of free electrons per $m^3$ of wire.] But, fundamentally, it's Lorentz forces that give rise to the force on a coil or any other configuration of currents in a magnetic field.
For two magnets, the repulsion experienced by one magnet, A, is due to the magnetic field set up by the other magnet, B. How does A get to experience this force? We can imagine the electrons in A's atoms to be moving in circles and therefore to be like little current-carrying coils (see previous paragraph). The electrons, being bound to atoms, pass on these forces to the magnet as a whole. This is essentially Ampère's theory from the 1820s.  
Ampère's theory, in which the force on a magnet is clearly derived from the Lorentz force, doesn't really carry over to the more correct quantum mechanical (or, better still, quantum electrodynamic) treatment, in which electron spin plays a key role. On account of their spin, electrons do experience a force in a non-uniform magnetic field, but whether or not we should regard this as a case of the Lorentz force in action, I don't know. Perhaps someone else can explain.  
