the flow of a linear system
Imagine the state of your system as a single point living in a space — for two unknowns, a point in the plane; for n unknowns, a point in n-dimensional space. As time runs, that point moves along a curve. Now do this for every possible starting point at once, and you have a whole moving field of points, like a fluid in motion. The flow is the rule that says, for each starting point and each elapsed time t, exactly where that point has been carried. It is the system viewed as motion rather than as an equation.
For the linear system x' = A x the flow is given by the matrix exponential: the operator that takes a starting state x0 and returns its position after time t is multiplication by e^(At). Write it as the map x0 maps to e^(At) x0. This map is linear (it is just a matrix), it does nothing at t = 0 (e^(A0) = I leaves every point fixed), and running for time t then a further time s is the same as running for t + s in one go, because e^(As) e^(At) = e^(A(t+s)). So the family of maps for all t fits together consistently — that consistency is exactly the semigroup property, and it is what makes 'flow' the right word.
Thinking of the solution as a flow, rather than as a formula for one trajectory, is the doorway to the qualitative theory. The flow can stretch space along some directions and squeeze it along others (the eigenvalues set the rates), rotate it (complex eigenvalues), or hold parts of it fixed (zero eigenvalues). Equilibria are points the flow never moves; stability is whether nearby points are pulled in or pushed out. All of phase-portrait analysis is reading the geometry of this flow.
For A = [-1, 0; 0, -2], the flow e^(At) = [e^(-t), 0; 0, e^(-2t)] shrinks every starting point toward the origin, faster along the second axis. Release a cloud of points and watch it collapse to the origin — that collapsing motion is the flow, and the origin (a point the flow never moves) is a stable equilibrium, a sink.
The flow is the whole moving field, not a single path; one trajectory is just what one chosen point does under it.
The matrix exponential gives a flow valid for all t, forward and backward, only because A is constant; for nonlinear systems a flow exists but may run only on a limited time interval before a solution blows up.