A rectifier is an electrical device that converts alternating current (AC) to direct current (DC), a process known as rectification. Rectifiers have many uses including as components of power supplies and as detectors of radio signals. Rectifiers may be made of solid state diodes, vacuum tube diodes, mercury arc valves, and other components.
Thanks Wiki
In layman's terms though, the socket on your wall supplies anywhere from 110 to 240 Volts AC (depending on your location). Your guitar amp, solid state or tubes, needs DC voltage to operate. Therefore the power supply of any guitar amplifier has the job of changing that 240V AC to whatever DC voltage your amp needs. (From 30-40V for a Solid State amp to several hundred volts for a tube amp). As you see from the Wiki definition, converting AC to DC is what rectifiers do, so you'll find one in every guitar amp ever made.
Most modern tube amps use a Solid State Rectifier (Silicon diodes) for a number of reasons, not least of which is cost and reliability. However you'll find tube rectifiers on vintage amps because silicon rectifier technology wasn't around back then. These days you'll find tube rectifiers in boutique tube amps because some people like the sound of an amp with a tube rectifier especially little amps like reproductions of the Fender Champ.
Now to answer the specific Mesa question..
I'm no boogie expert so I'm sure someone will correct me, but as I understand it, the dual recto's had (not surprisingly) two rectifiers, a Tube and Silicon and you could switch between them to get the sound you wanted. I believe the triple rec's include both tube and silicon rectifiers as well as a some kind of "bold/sponge" switch as well (but I have No Idea what that does....)
Overall though it's just a brand name. There are LOADS more tonal and usability changes between the dual recto and triple recto than type of power supply.
Oops one more thing...
People like tube recto's in small amps because of a phenomena known as "Sag". Basically tubes do a terrible job at rectification so as you wind up the volume and hit big bass notes, your power supply voltage will "sag" or drop. In a tube amp this (i think) equates to more compression which some people dig. Note that for this to occur you need to be -cranking- that amp. Totally reasonable in a 5W champ, aural suicide with a 150W tube boogie. Silicon is FAR better at it's job so you get no sag, potential for more volume given the same circuit and of course a different tone when you -crank- your amp.