The Michelson and Morley experiment was a momentous event in the lead up to special relativity. In this section, we'll explain Michelson and Morley's search for experimental evidence of a luminiferous ether. (Spoiler...they didn't find it. And that blew people's minds.)
In a Newtonian world we assume that the passage of time or lengths in space are absolute--that regardless of one's inertial frame of reference, everyone would agree that the same time passed in every frame of reference and that something of a certain length in one frame would be the same length in other frames. This leads to the speed of light needing to be relative depending on the frame of reference which contradicts observation. We reconcile observations by loosening our assumptions about absolute space and time. We explore this through Minkowski spacetime diagrams.
We'll now put some algebraic meat on our special relativity bone by introducing ourselves to the Lorentz transformation. Interestingly, Lorentz came up with them before Einstein introduced Special Relativity as a nice way to reconcile some observations, but special relativity gave them real conceptual meat. The Lorentz transformation is going to allow us to calculate the spacetime coordinates in different frames of reference!
You're traveling toward me at half the speed of light. My friend is traveling away from me at half the speed of light. From either of your points-of-view, how quickly are you approaching each other? Things are going to get even more fun now. We're going to think about relative velocities in different frames of reference.