Link popularity is essentially the measure of how many links point towards your site, and the quality of these links. Naturally, the more links to your site, the more popular your site is. Note that link popularity is different from PageRank. PageRank is just Google's way of telling users how relevant the site is to its field, and how important the site is in general.
Obviously, being popular on the Internet has many, many benefits. More people will know about your site, and this is good for traffic and sales. However, make sure you are popular for the right reasons. Quality sites will link to your site if people find it useful and is generally very pleasing. Likewise, the same sites might link to you if your site is extremely bad, to show people an example of a bad site or to tell them to stay away from it.
So, to be popular on the Internet you must have a quality site yourself. But it doesn't help to speed up the linking process a bit by yourself. Go out and exchange links with relevant sites. Of course you should be aiming for the quality sites, but don't rush. Start with sites that are roughly your level in terms of age and popularity, then work your way upwards. Remember to tell people what your desired anchor text is to rank better.
Or, if link exchanging is not your thing, maybe you should focus on the content of your site. Create a site that is interesting, something people would bother with. People are busy, and if your site offers nothing special it'll be more likely than not ignored. If you're going with this strategy, it is imperative to make use of social bookmarking sites to help users stumble into your site.
Look around for sites that are similar to yours that rank highly on search engines. Study them. Find out what they have that you don't, and get as close to imitating them as possible without blatantly copying them (because this will do more harm than good). In other words, find out what their strategy is, the structure of their site, and the overall quality, and try to match this quality or even outdo it.
Of course, the ideal way to go about this is to try everything mentioned here. If you really don't have the time, you might consider hiring professional SEOs. Good luck!
