The dome on the far left is the one that housed the first laser guide star on Mauna Kea... it was a monster dye laser that consumed ~30kW of electrical power to produce a 10-15 Watt laser beam of 589nm wavelength light tuned to the sodium atomic resonance.

My old laser has long since been replaced by a far more efficient solid state laser as technology has rapidly advanced.

The beam would be aimed at a point somewhere in the field of view of the telescope, close to the region of interest, since the amount of correction falls off as you look further away from the guide star. For this reason some observatories transmit a 'constellation' of guide stars, 4 beams that provide references across the field of view.