So, let's talk about gravity. Gravity is a force between objects which have a certain kind of property, let's call it property X. Gravity has a precise quantitative relationship with property X, the more property X an object has the more gravitational force it exerts.
What is "property X"? It's energy. And here is where things get a bit complex, because most people would instead have said that "mass" is property X. The problem with that is mass becomes variable depending on the reference frame, and it turns out to add a lot of excessive complexity to discussing things, especially when precision is required.
So you could imagine talking about mass as the equivalent of energy, which is typically an accurate viewpoint, and then you get to the idea of "relativistic mass". Which is the adjusted "property X" value of an object which might be traveling at relativistic speeds in a given reference frame.
Relativistic mass, or property X, can be a helpful mental model in some ways, and in normal uses of English it's often a more useful way of thinking about things. But it's also problematic because it's ambiguous.
This has led to a bit of an impedance mismatch between the way physicists talk about relativistic effects and the ways that it's more natural to talk about such things in plain English. In English "property X" is mass, but in physics it's actually energy, and it's difficult to get people to fully grok the intimate relationship between energy and mass.
Physically, mass is just a special name for invariant, or rest, energy, the energy of an object in the reference frame where the object is stationary. It's all energy, but it's important to separate out rest-energy vs. energy in a given reference frame, and so forth.
What's the backslash, is that m times u or μ, is that exponentiation, which are vectors and which are scalars?
mu is being used as an index (mu=0,1,2,3) on the components of the vectors p and u, m is a scalar representing rest mass.
This notation can be naturally extended to tensor products of vectors in the tangential and co-tangential spaces to the base manifold that is spacetime (simply called "tensors" by physicists): https://en.wikipedia.org/wiki/Einstein_notation
The concept of dynamic mass is motivated by wanting to continue to write the previously known three-momentum as p = m v, which does not conform to special relativity, hence the definition of mass is changed. However, in a formula as basic as F = m a (F and a being vectors), substituting the dynamic mass for m does not yield correct results because in general, under special relativity, F and a do not even have to be parallel.
Modern formulations of dynamics in special relativity use the more intuitive invariant mass, and three-momentum is written as p = m gamma v, where gamma is the factor previously included in m_r. This p is now the spacial components of four-momentum p^\mu = m u^\mu, where m is the invariant mass and u is the relativistic four-velocity of the moving object.