The lighter your load, the faster your powers cool down. Different classes have different max carry capacity, and you can boost it through upgrading their passive class ability. The biggest bonus to power regen speed you can get is +200%, meaning a 9 sec cooldown will be reduced to only 3 seconds. Soldiers and Vanguards, I believe, have the highest weight capacity, while Adepts and Engineers have the least.
As a Sentinel, I've managed to carry along a shotgun, an assault rifle and an SMG (Just for kicks, as they are pretty much weightless with the proper mods and upgrades) while maintaining a respectable power regen bonus (+180% or so).
Squadmates are unaffected by weapon weight, so feel free to pack them with heavier weapon variants.
Sniper rifles are the heaviest weapons, followed by shotguns and assault rifles. Pistols and SMGs are fairly light with a few exceptions. SMG weight can be reduced to nearly nothing with appropriate mods and upgrades. There are a few light assault rifles, which also tend to be fairly accurate and stable, making them ideal for Shepards.