The way I see it, there are 3 distinct types in use by the GTVA. My decidedly non-canon interpretation:
Green: Terran beam weapons which work by bleeding off plasma from the ship's reactor. These have the advantages of being relativley small, allowing a given vessel to carry more of them, and they can be 'overcharged' by forcing greater amounts of plamsa through the emitters. This comes with the cost of increased thermal load on the cannons, and can also damage the reactors themselves if too much plasma is drained.
Yellow: Vasudan beam weapons which utilise directed photon beams. These are more powerful than the green Terran beams, but require a dedicated reactor to power them. As a result of this increased bulk, the weapons are typcially mounted in smaller numbers, and not at all on cruisers. We've not seen if these can be 'overcharged'.
Blue: Ion beams shared by both races. These are very energy efficient and have a low thermal load on the emitters, allowing for rapid refiring and continual operation, but do very little damage. Because they can draw power from the carrying vessel's standard energy grid, practically any vessel of cruiser tonnage or above can mount worrying numbers of these weapons.
As for the Shivans, I like to think of their beams as being some kind of graser, if only due to rule of cool.