Response spectrum, explained without the math first
A response spectrum looks like a dense curve buried in code clauses, but the physical idea behind it is a simple thought experiment: how hard would a whole family of different buildings shake, each swaying at its own natural rhythm, during the same earthquake?
Imagine lining up a whole row of simple, single-mass pendulum-like structures, each with a different natural period — some very stiff and quick to sway back and forth, some very flexible and slow — and subjecting every one of them to the exact same recorded ground-shaking history. Some of those imaginary structures would end up shaking violently; others, barely at all — not because the ground motion was different, but because how strongly a structure responds to a given earthquake depends heavily on how its own natural period lines up with the frequencies actually present in that particular shaking. Plot the peak response of every one of those imaginary structures against its own period, and that curve is a response spectrum: a period-by-period map of how hard a structure swaying at any given rhythm would respond to this earthquake.
From one earthquake's curve to a smooth design spectrum
A response spectrum built from a single real recorded earthquake is jagged and specific to that one event — not something you'd want to design a real building against, since the next earthquake at that same site would trace a different jagged curve. Design codes instead publish a smoothed, generalized spectrum shape, built by studying many recorded ground motions and site conditions, meant to represent a reasonable envelope of what a building at a given site, on a given soil type, might realistically experience — parameterized by the site's ground type and its seismic hazard level (peak ground acceleration), not tied to any one specific historical earthquake record.
The spectrum a code actually publishes comes in two related forms: the elastic response spectrum describes how a structure would respond if it never yielded at all (the reference the whole method is built from), while the design spectrum divides that elastic demand down by the ductility-related reduction factor covered in the previous entry (q or R) to get the reduced force level a real, appropriately detailed structure is actually checked against. Seismic Base Shear builds and plots this full spectrum curve for the code and ground type you select, and reads off the specific design spectral acceleration at your building's own computed fundamental period to get its base shear (see the related tool below) — the spectrum isn't just a formality, it's the whole physical link between what the ground did and how hard this specific building actually gets shaken.