Hello,
may I ask the gurus a question please.

What does a SWR meter actually display?

effective power
reactive power
apparent power

What does it display in the 1 : 1 case with optimal impedance matching
between transmitter, cable and antenna, I guess it's effective power,
but what does it display when matching is off, say, 1 : 5
or even worse, when the mismatch SWR is 1 : 10 or greater?
There are two pointers in the instrument, forward and reverse,
how do they add up? Why do forward power and reverse power both
increase in their values when the antenna is strongly mismatched,
but the displayed SWR remains the same?

http://en.wikipedia.org/wiki/SWR_meter gives a decent starting view of
how a typical SWR meter works.

To a first approximation: an SWR meter is measuring the magnitude of
the forward wave (transmitter - antenna) and reverse wave (antenna -
transmitter) RF. The latter (the reverse wave) is usually a portion
of the forward wave which has been reflected back towards the
transmitter when it encountered an impedance change.

Again, to a first approximation, the "effective" power being
transmitted is equal to the power in the forward wave *minus* the
power in the reverse wave. So, you don't want to "add up" the two
powers you see displayed... you want to subtract one from the other,
to see what's actually being radiated.

This is only an approximation due to the presence of losses in the
transmission line, between the SWR meter and the antenna.

If there's enough loss in the transmission line, the reflected wave
will be mostly dissipated in the line as heat... little of it will
reach the SWR meter. In this case, even if there's a horrible
mismatch at the antenna (e.g. the antenna is open or shorted) the SWR
meter won't see a significant reflected wave, and will indicate a good
match (close to 1:1).