Here's a Wikipedia article about the efficiency ratings.
The efficiency of a computer power supply is its output power divided by its input power; the remaining input power is converted into heat. For instance, a 600-watt power supply with 60% efficiency running at full load would draw 1000 W from the mains and would therefore waste 400 W as heat. On the other hand, a 600-watt power supply with 80% efficiency running at full load would draw 750 W from the mains and would therefore waste only 150 W as heat.
As others have noted, the power supply can/should always be able to supply its rated power. But it draws even more power from the wall than that.
For easy math, let's assume a 100W power supply. A 100% efficient PSU would draw 100W from the wall and provide 100W to the system. A 50% efficient PSU would still provide 100W to the system, but it would draw 100/0.5 = 200W from the wall.
The 80 Plus certification will certify different levels of efficiency when the PSU is providing different percentages of its maximum output. For example, according to that Wikipedia link, an 80 Plus
Bronze PSU will be 82% efficient when outputting 20% of its maximum load, 85% efficient when outputting 50% of its maximum load, and 82% efficiency when outputting 100% of its maximum load. The 80 Plus
Bronze standard specifies no level of efficiency at 10% load. Meanwhile, an 80 Plus
Titanium PSU will be 90% efficient at 10% load, 92% efficient at 20% load, 94% efficient at 50% load, and 92% efficient at 100% load.
There are two benefits of higher efficiency: less power draw (so a lower energy bill), and less waste heat (so quieter fans).
I suppose a more efficient PSU might live longer too, since it has less heat. On the other hand, if it takes advantage of the reduced heat by running its fans slower and quieter, then it might end up having just as much heat, so maybe it won't live longer. I'm not sure.
Personally, I use the 80 Plus certification as an extra check for overall PSU quality. I figure, if the PSU is high-enough quality to be independently certified as highly efficient, then it's probably very high quality in other ways too. That's probably not a 100% certain way of testing PSU quality, but I figure, it's one more thing.