I'm really surprised you wouldn't know this with all the math on your webpage.
I'll do a simple example. Suppose you have a coin that you believe is not fair. If it was fair, it would heads 50% of the time, and tails 50% of the time. Of course, since you cannot do infinite trials, you will have to do a limited set of trials. Suppose you flip it 100 times. Obviously this sample size is pretty bad. It may end up heads 52 times and still be a fair coin. But suppose it was heads 60 times. Is this coin fair?
At first glance, it seems as if it is not. So this is where the test comes into play. The distribution of a random coin should match a binomial distribution, which has tables. These tables (which can be calculated as well), will tell you how many trials out of a fair distribution will score above (or below) a certain point. The binomial distribution looks very similar to a normal curve for large sample sizes.
Binomial Distribution
Plug in the parameters here, and there is a 1.8% chance that a fair coin will score 60 or more heads. So this test fails. In other words, we cannot prove the coin is not fair. However, this does NOT prove the coin is fair.
Unfortunately, you cannot really prove Party Poker is fair, you can only prove it was not fair. But, you could use these numbers to say with a confidence that a certain card does not come up with an unusual amount of times.
It's been a while since I've done this, but anything I screwed up, feel free to correct me.