the spacetime interval
Two observers flying past each other disagree about almost everything: how far apart two events are, how much time separated them, even their order. Yet there is one number about a pair of events that they always compute to be identical — the spacetime interval. It is relativity's bedrock invariant, the 'distance' of the geometry of spacetime.
The interval between two nearby events is ds^2 = eta_munu dx^mu dx^nu = c^2 dt^2 - dx^2 - dy^2 - dz^2 (signature +,-,-,-). Its sign classifies the separation: ds^2 > 0 time-like (a massive particle can travel between the events, and proper time elapses, c^2 dtau^2 = ds^2); ds^2 = 0 light-like/null (only light connects them); ds^2 < 0 space-like (no signal can, and simultaneity is observer-dependent). Under a Lorentz transformation the individual dt and dx change, but the combination ds^2 is invariant — that invariance is the defining property.
The interval is the relativistic replacement for 'elapsed time' and 'distance' as separate absolutes; both are demoted to frame-dependent projections of the single invariant. You meet it when deriving time dilation and length contraction, defining proper time along a worldline (tau = integral of ds/c), and building every four-vector. Caveat: because the metric is indefinite, ds^2 can be negative, so ds itself is imaginary for space-like separations — which is why one usually quotes ds^2, or uses proper time for time-like and proper length for space-like separations.
A muon created in the upper atmosphere and decaying at ground level: in Earth's frame it travels, say, 10 km in 34 microseconds; in the muon's own frame it travels zero distance in its proper time (about 2.2 microseconds). Both frames get the same ds^2 = c^2 dtau^2, which is how the muon reaches the ground despite its short lab lifetime.
The invariant interval reconciles the two frames' wildly different times and distances.
Do not read ds^2 as a squared real number that must be positive — the Minkowski 'metric' is indefinite. A 'zero interval' between distinct events (light-like) is perfectly physical and does not mean the events coincide.