A jacket is said to be waterproof when it, quite simply, keeps water from getting to the wearer’s body and soaking it.
There are different standards for measuring waterproofness; one way is to measure the resistance of the fabric in terms of PSI. Government standards dictate that a fabric is waterproof if it has a rating of 25 PSI or more. Since rain is somewhere in the neighborhood of 7 PSI, there’s a good chance you won’t get wet in a rainstorm if the fabric is of 25 PSI or more. And it follows that a stronger, more durable fabric will be better able to stand up to stronger rains, etc. For instance, Sierra Designs makes a jacket called the Neutron, which is fully waterproof, but weighs about 4 ounces (!). Arc’Teryx makes a jacket called the Alpha SV. It weighs nearly 20 ounces, but it’s constructed of high-quality Gore-Tex, and so is going to stand up to the elements a lot better than a jacket 1/5 its weight.
Another way to calculate waterproofness is with a number in the thousands. This number measures how many thousands of millimeters of water a fabric can tolerate, in a 24-hour period, before it succumbs to the pressure. For example, one of my jackets, a Gore-Tex shell, is rated at 30,000mm/24 hours. That means that it can withstand a water pressure of 30 meters within 24 hours before it “leaks.” Look at it this way. Pretend that there is a fabric stretched tightly, so that it’s completely flat. There is a tube on top of it, measured in millimeters. My Gore-Tex shell can withstand the pressure from a tube of water 30,000 millimeters high, within 24 hours. That’s going to be on the upper end of the spectrum, for waterproof integrity. On the other end, you might have a jacket that is rated at only 3000mm/24 hours. The “tube” is going to be 1/10 the height; it’ll succumb to water pressure a lot earlier than Gore-Tex. Of course, this comparison is not quite fair; these numbers are in-lab, and don’t always correspond to real-world performance; additionally, different companies have different ways of testing fabrics. But generally, the higher the number, the more waterproof the jacket is.
In the event that you find a jacket using one measure of pressure, and want to know what it is in terms of the other measure, there's a quick and dirty approximation you can use to convert one to the other. Suppose that a jacket is rated at 30 PSI. You simply halve the PSI, then multiply it by 1000, to get the measure in terms of mm/24 hours. Back to the 30 PSI: 30 divided by 2 equals 15, which is then multiplied by 1000, to get 15,000mm/24 hours. This is a rough estimate, but it's helpful in estimating similar water pressures.