I was told that Lorentz' force is given by
$${\bf F}= q{\bf v} \times {\bf B}.$$
But I have read that it is given by $${\bf F}= \frac{q}{c}{\bf v} \times {\bf B}.$$
Why have these two relations different forms if they represent the same force? Thanks for any help!
Both relations you mention are completely equivalent, the only difference being the system of units in which they are expressed. Every system of units $A$ is consistent with any other system of units $B$ as long as you yourself are consistent in their usage and correctly transform everything between $A$ and $B$ when desired. So the factor of $c$ does not constitute a conflict, which can be seen when the equations are transformed. This is shown in this useful section on the wikipage of the Lorentz force.
Looking at the full Lorentz force expression (of which your expressions are a special case with no electric field), the first one you mention, $$\vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right),$$ is expressed in SI units.$^1$ The second relation, $$\vec{F} = q\left(\vec{E} + \frac{1}{c}\vec{v}\times\vec{B}\right),$$ is expressed in Gaussian units. So both relations are equally valid, as long as you use the correct expression consistent with any other expressions - meaning you should at all times stay within the same system of units. Consistency is key.
The SI way of writing these kinds of electromagnetic expressions is steadily gaining popularity and most new books adopt this system of units, but Gaussian units have long been dominant. You will therefore mostly see those units used in older books. Also in literature these units are still widely used.
Transformations between Gaussian units and SI units require some extra care since - as you may have noticed on the wikipage I linked - they are not simple (dimensionless) rescaling transformations. As a consequence it is possible for a dimensionless equation in SI units (Gaussian units) to yield a non-dimensionless equation when transformed into Gaussian units (SI units).
One example is when we consider Gauss's law in Gaussian units divided by the free charge density: $$(1/\rho)\vec{\nabla}\cdot\vec{E} = 4\pi.$$ The quantity on the left-hand side is dimensionless in Gaussian units, but not in SI units, where it is $$(1/\rho)\vec{\nabla}\cdot\vec{E} = 1/\epsilon_0.$$ So you have to watch out for that when transforming your equations. Dimensional analysis may therefore also yield seemingly different or contradicting results, but there is no problem if you remember the conventional differences and, again, stay consistent.
$^1$ Note that the expression for the Lorentz force also looks like this in natural units, which is another widely used system of units. Here the units are chosen such that certain natural constants such as the speed of light $c$ have a numerical value of 1. It is then common practice to omit those constants from all equations, for sake of simplicity.
