For a linear time-invariant system, the impulse response h(t) or h[n] fully characterizes the system's input-output relationship. Any output can be computed as the convolution of the input with the impulse response, making the impulse response the most fundamental descriptor of an LTI system.
From your study of system properties, you know that a linear system obeys superposition: if input x₁ produces output y₁ and input x₂ produces y₂, then αx₁ + βx₂ produces αy₁ + βy₂ for any constants α, β. A time-invariant system has behavior that doesn't change over time: if x(t) produces y(t), then x(t−t₀) produces y(t−t₀). These two properties together — LTI — are a remarkably powerful combination because they allow any input-output relationship to be captured by a single function: the impulse response.
The logic works as follows. The impulse δ(t) is the idealized "spike" signal from your elementary signals prerequisite — infinite height, zero width, unit area. Apply it to an LTI system and record the output: that output is h(t), the impulse response. Now consider any arbitrary input x(t). You can decompose it as a sum (really, integral) of scaled, shifted impulses: x(t) = ∫ x(τ)·δ(t−τ) dτ. By linearity, the output is the sum of the responses to each scaled impulse; by time-invariance, the response to a shifted impulse δ(t−τ) is a shifted version of h, namely h(t−τ). Combining: y(t) = ∫ x(τ)·h(t−τ) dτ. This integral is convolution, written y = x * h. Everything the system can do to any input is encoded in h.
To build intuition, think of a simple echo system: if you clap in a concert hall, the hall responds with a decaying sequence of echoes. That decay pattern is the impulse response h(t) of the hall. Any sound x(t) played in the hall produces the output y(t) = x * h — the dry signal convolved with the echo pattern. Audio engineers record impulse responses of real spaces (by firing a starter pistol) and then convolve any dry recording with that impulse response to digitally place the recording in that acoustic environment. The system is the concert hall; its complete behavior is captured by one measurement.
In discrete time the same logic applies exactly, with sums replacing integrals: y[n] = Σ x[k]·h[n−k]. A digital filter is completely defined by its impulse response sequence h[n]. An FIR (finite impulse response) filter has h[n] that is nonzero for only finitely many values of n; an IIR (infinite impulse response) filter has a response that continues indefinitely, typically implemented with feedback. The connection to frequency domain analysis is direct: the Fourier transform of h(t) is the transfer function H(f), which tells you how the system scales and phase-shifts each frequency component of the input. This connects the time-domain convolution picture to the frequency-domain multiplication picture that will be essential for pole-zero analysis in future topics.