spacetime interval
The spacetime interval is relativity's answer to a hard question: with everyone disagreeing about distances and durations, is there anything two events have between them that all observers agree on? The answer is yes, and it is the interval s^2 = (c t)^2 - x^2 - y^2 - z^2, built by subtracting the squared space separation from the squared time separation. Different observers will measure different t and different x, but they always compute the same s^2 — it is invariant.
The crucial feature is the minus sign. In ordinary geometry you only add squares, so a distance is always positive. Here the time and space pieces fight each other, so s^2 can come out positive, negative, or exactly zero, and that sign carries deep meaning. If time wins (s^2 > 0) the events are 'timelike' separated — one can cause the other, and a clock can travel between them. If space wins (s^2 < 0) they are 'spacelike' — too far apart in space to influence each other. If they exactly balance (s^2 = 0) they are 'lightlike', joined only by a ray of light.
A useful everyday image: think of the interval as a kind of cosmic odometer reading that two cars agree on even though their dashboards show different distances and trip times. For timelike-separated events, that invariant reading is literally the proper time a clock would record travelling directly between them. So the abstract-looking formula is really just measuring lived time, and that is why every observer must agree on it — they cannot disagree about how much a single clock ticked.
Time and space subtract, not add; the sign of s^2 classifies the pair as timelike, lightlike, or spacelike.
Sign conventions vary: some books write s^2 = x^2 + y^2 + z^2 - (c t)^2 instead, flipping which separations count as positive. The physics is identical — only the bookkeeping changes, so always check a text's convention before comparing signs.