Convolution is basically: the output of a system at time=t is the sum of the system's affect on all (or for causal systems, all previous) inputs.
So if the input is three pulses -- one at t=0, one at t=1, one at t=2, then the output of the system at t=10 seconds is the effect the first pulse has after 10 second plus the effect of the second pulse after 9 seconds plus the effect the third pulse has after 8 seconds.
This is also why the argument to the impulse response has the subtraction.
"mixing" is sometimes used to mean simple multiplication. Although some fields use it to mean addition.