There are two types of mass; rest mass and relativistic mass. The rest mass does not change with velocity or reference, whereas the relativistic mass will change with velocity and reference. The rest mass of the proton will be the same in all references. However, the relativistic mass of a proton can be different for different references.

It would take infinite energy to move one unit of rest mass, the speed of light. The rest mass does not become infinite, at the speed of light, or else you would create a black hole. Rest mass is impacted by gravity. What changes is the relativistic mass. Relativistic mass is a mass equivalent, associated with kinetic energy, where kinetic energy is being modified by changes in time and space. Relativistic mass has a connection to time.

As an example, say we have two references, A and B, side-by-side; hypothetical. Reference B is moving faster than reference A, such that time in reference B, is running slower due to special relativity. What I will do is dribble a basketball in reference A, and then stick why hand into reference B, while still dribbling the ball. Because time is running slower in B, the dribbling ball appears to slow down, relative to A. To get the ball to move as fast in B as it did in A, I will need to apply more force; accelerate the ball.

After applying the force and letting the ball hit the ground and rebound, the ball is now going the correct speed. However, when the ball hits my hand, what I notice is the ball now appears to have more inertia than it did in A, at that same velocity. This extra apparent inertia, is due to the extra force I added, to compensate for the slower time. The rest mass has not changed, but the ball feels heavier, due to the extra inertia. That extra inertia is the relativistic mass. The rest mass does not change, but the extra apparent mass, as inferred by more inertia, is connected to the added force needed to normalize the differences in time.

In terms of the directional nature of relativistic mass, the apparent inertia changes in the direction of the force.