Link Density refers to the total number of links on a page. Google Webmaster Guidelines recommend keeping the number of links on any one page to a "reasonable number" (they suggest under 100). Pages that are very link heavy can sometimes be indicative of a Bad Neighborhood (ie. a blog that has been heavily spammed, or a link farm), although obviously this is not always the case.
The Bad Neighborhood Warning icon indicates that somewhere on the page there was a link that had anchor text that could be indicative of industries or fields which tend to have a high incidence of search engine spamming, or may not be considered "family friendly". Industries which fall under these categories would include areas such as Adult Entertainment, Online Gambling, and Pharmaceutical.
The Page Checks Clean icon indicates that no links on this page that were found had signs of linking to potentially risky websites, based solely on anchor text evaluation.
The Redirected icon indicates that the scanner encountered a server side redirect response when querying the page. The redirect was either a 301 Moved Permanently or a 302 Found (sometimes referred to as a "Moved Temporarily" response), and should not be confused with meta-refresh or Javascript style redirects.
The 403 Forbidden icon means that the server would not allow our bot to spider the page in question. It may mean that the page itself is restricted from public access. This is not an indicator of a Bad Neighborhood in any way, it just means that we were unable to spider and analyze the page at this time.
The 404 Not Found icon means that the page that we attempted to spider does not exist. While missing pages in and of themselves are not signs of a Bad Neighborhood, it may mean that the site linking to the missing page is not updated as frequently as it should be. The general consensus is also that the search engines may determine that a site that leads to many 404's should have a lower trust factor.
The Blogspam icon, when displayed, indicates that the domain that the page exists on has a high occurrence of links that appear to be from blogs. While this is normal in many cases, such as with forums or other blogs, for many commercial websites this could indicate an attempt to increase search engine rankings through blog comment spamming.