Google’s original link messages as Matt refers to them are known by many website owners who used black hat techniques in the past and also those who didn’t, or didn’t realize their SEO company had.
These messages are notifications of unnatural link patterns and usually are sent when a site has many of these types of inbound links or a few in particular from link networks that Google has identified as being especially spammy. The action you should take if you get these original messages is quite clear, get rid of as many of the links as possible and then submit your site for reconsideration.
Matt Cutts though has clarified Google’s position on sites that charge to have links taken down or simply won’t take them down, you should it seems either do reconsideration request with the details, mention it in the Google webmaster forum an hope its picked up or do a spam report: how long it will take to see your trust return is unclear.
With the original link messages you probably had your Google trust level drop significantly or you could have had it disappear completely. With the new link messages you will get told of less severe cases of unnatural linking patterns and of individual links that Google would like to see gone. These links don’t affect your sites overall trust but trust of these links will be low and potentially the trust of the specific page that they point to will fall. They are then worth getting rid of still but it isn’t as urgent; of course if Google keep finding these links it could lead to a more major drop in trust so it is a good idea to deal with these new link messages as you get them.
What kind of inbound links are they then that Google is likely to be notifying you of in these new messages?
One type is widget links, those with keyword rich anchor text, which Matt Cutts has spoken about before, will it seems be seen as spam and you should now be working on building inbound links with anchor text that has different terms each time.
Then there are paid links, Google identify sites that sell links in a few ways, sometimes they will suspect them because of where the links are placed and when the link is on the same place on each page, Google also identify pages where links are advertised for sale including link directories that only offer paid for inclusion and ignore these inbound links in most cases.
Google can also identify blog comment spam where the same messages are posted on multiple sites or where they have nothing to do with the blog post and are stuffed with keywords and these are another type of link they may send this message for.
As well as removing these links if possible it is important to check what the problem with the link was and track where they came from: if your SEO or people in your company are creating these links they need to be stopped.