When a mining ASIC claims to run at 1 MH/s or 10MH/s or even 1000 GH/s, at what difficulty

10 MHash/s means that the chip can do 10 million hashes per second, regardless of what the difficulty is. The difficulty changes how many blocks you find, but it doesn't change how many hashes the chip performs.

and with what algorithm is that measured against?

It'll usually say what algorithms it's capable of mining, but if it doesn't, you can usually assume it's SHA256. There are a couple of odd ducks, like hybrid ASICs that can mine multiple algorithms at once; those are actually two different mining circuits put on the same chip.

cgminer shows the ASIC running at the hash rate claimed. But when mining in a pool, the pool shows a hash rate of roughly 1/3 what cgminer says.

There are three possible explanations:

• The chip is lying about how many hashes it's doing, but the pool can't be cheated in this manner, because it relies on shares. (unlikely)
• You're failing to submit otherwise valid shares, because of some software misconfiguration. (slightly more likely)
• You've been unluckly, and have gotten less shares than average. This will even out over time. (most likely)