In a
fluid flow a streakline is the
curve occupied by fluid
particles that have passed through some fixed
point x0. Looking at streaklines is a way to visualise a flow. In a steady flow the streaklines will coincide with the
streamlines and particle
paths, but this is not true in general for non-steady flows.
To compute what the streakline originating from x0 looks like at time t0 in a flow with velocity field u(x, t) you simply find the location of the fluid particle that was at x0 at time t' for all t' < t0. This is done by integrating the differential equation dx/dt = u with initial conditions x(t') = x0, which can take some effort.
Streaklines are not really the best way to visualise a flow. Rather their usefulness lies in the fact that in an experiment they are very easy to exhibit. All you have to do is to place somethings that emits some sort of dye at x0 and the streakline will be visible as a dyed streak.