On a tube amp, the gain knobs affect how hard the tubes are being driven in a particular section, so the more you turn the gain up, the more the amp will go into overdrive. Some tube amps have pre-gain to drive the preamp stage and post-gain to drive the power amp (like the Peavey Blues Classic, for eg), while the master volume determines the overall signal sent to drive the speakers. Amps that don't have post-gain control can still achieve that power amp overdrive when run at high volumes.
I don't think you'll want the technical explanation for what a tone knob actually does (controlling a variable shelving filter with a first order response), so let's go with the musical one.
The natural pickup sound has a strong low-mid emphasis and little high frequency response, which is actually a muddy and muffled sound, and requires correction and compensation for it to sound musically pleasant. The most common tone control in guitar amplifiers is a design called a tone stack, so called because the bass, middle and treble controls are electronically connected in series, usually drawn as a vertical stack. This design has the advantage of inherent middle-cut, serving a dual purpose of general tone balance correction along with tone control.
My favorite clean setting on an amp usually start with Bass/Mid/Treble on 6/6/3 respectively. Bass on 6 provides body without sounding boomy, Mid on 6 helps your guitar cut through the mix and highlights your pickups' character (bearing in mind that pickups are naturally middy), and Treble on 3 leaves out the harsher frequencies that might otherwise cause ears to bleed in the audience.
Presence boosts the upper frequencies above the normal treble control range for added high-end "sparkle". It works a little differently from the other tone knobs (It's a lowpass filter inside a global negative feedback loop. By decreasing the amount of high frequencies that are fed back, the high frequencies at the output of the amplifier are boosted).
All-in-all, crank it up, and have fun.