Google uses link popularity more than any other engine, and it also provides the best tools for measuring your link popularity. Also Google is the most popular search engine in the world, and link popularity matters more than any other criteria when ranking high in it. As such I've decided to devote an entire section just on how Google uses link popularity. Since Google also pioneered the modern link popularity approach what you learn off Google can be applied with success to other engines that use link popularity (in all honesty most major search engines have more or less copied Googles ranking algorithm).
The first thing I need to mention is the Google Toolbar. Google offers a plugin for Internet Explorer that provides some various features. These include a bar which displays the current URL's page rank, as well as easy one click ability to check which sites link to the current page. I cannot recommend enough this toolbar, I consider it a must for any webmaster.
The first thing you will notice in the toolbar is a green bar that indicates a PageRank level. To find the specific numerical value associated with the green bar you can hover the mouse over it. PageRank is Google's version of link popularity and takes the weight and context of incoming links into account. The amount of PageRank needed for each individual level increases exponentially. It is much easier to go from a rank of 3 to 4 than to go from 5 to 6. In fact one or two good links could very well give a page a rank of 4, however you might need a thousand to get a 6. Its very hard to figure out exactly what the formula is, but research done by myself points to the base being a 4 or a 5. So each level would be 4 or 5 times as hard to reach as the one previous to it.
It is important to note that the value given in the toolbar is not always accurate. It does not show the actual PageRank for the page you are currently visiting, merely an approximation that can be up to 4 months old, so take it with a grain of salt. Additionally, if you are visiting a page that Google has not yet spidered, but they have spidered the root domain of the site, then they will guess a PageRank based on the distance from the root of the site to your current position. This is only a guess and has no bearing on ranking. Once Google spiders the page they will assign an actual PageRank to it. This guessing behavior is the reason why some Geocities pages sometimes seem to have high PageRanks, the fact is they've simply not been spidered yet.