the time domain and the s-domain
Think of a problem stated in English and the very same problem translated into another language. The meaning is unchanged, but some sentences that were clumsy in one language become effortless in the other. The time domain and the s-domain are two such languages for describing the same physical situation, and the Laplace transform is the dictionary between them.
In the time domain you have a function f(t): the actual quantity — a current, a position, a temperature — as it unfolds second by second. Apply the Laplace transform and you land in the s-domain, holding a function F(s) of an abstract variable s. The two carry identical information: nothing is lost, you can always translate back with the inverse transform. What changes is the difficulty of the operations. In the time domain a derivative d/dt is a calculus operation; in the s-domain it becomes (roughly) multiplication by s. In the time domain solving an equation may require clever calculus; in the s-domain it is rearranging an algebraic formula.
The whole strategy of Laplace methods lives on this picture: take a hard problem stated in the time domain, walk through the dictionary into the s-domain where it becomes easy algebra, solve it there, then walk back. Engineers often think in the s-domain directly — concepts like poles and transfer functions are statements about F(s) — precisely because that domain exposes structure (stability, resonance) that is hidden in the wiggles of f(t).
A unit step turned on at t = 0 is the function f(t) = 1 for t >= 0 in the time domain. Its s-domain face is F(s) = 1/s. The same fact viewed two ways: 'the signal is constantly on' (time) and 'a 1/s pole at the origin' (s).
One signal, two equivalent descriptions linked by the transform.
The variable s has no direct physical meaning the way t (seconds) does — it is a bookkeeping variable. Resist trying to picture s as a time or a frequency; just treat the s-domain as the workspace where the algebra is easy.