SEO web hosting
Webmasters try to avoid being penalized for their search engine optimization attempts, by making it appear that everything is occurring naturally, especially the very important inbound links from other sites.
The goal is to make it look like a bunch of different, unrelated sites, are linking to your desired website. Often times the goal is extended to a whole group of websites, all working in concert to help each other win the search engine optimization race.
Even though the webmaster in fact controls all those websites, the relationship must be hidden. A first and most obvious way of doing that is to have unique or at least substantially different domain whois data. Domains owned by the same person, linking to each other, would be enough to raise suspicion. Using different whois privacy services can be a partial solution to that sort of problem.
The role of IPs
Each website in the end points to an IP, where it is hosted. In the background, all domain names actually point to an IP. Typically, a good number of websites will be sharing the same server, and the same IP. That's because most websites have limited needs, and it doesn't make sense to have them hosted on a dedicated server, nor on dedicated IPs.
Search engines are well aware of this, it is entirely expected and normal, so using shared hosting will not by itself be a worry for your SEO efforts at all. We actually have official confirmation of this policy from a Google employee.
However, you will often hear people swearing that having a dedicated IP bring you extra SEO points, or that it is safer to have a dedicated IP, so that you don't get penalized by unsavory websites you share the IP with. The only minor truth to sharing IPs has to do with email spam. If for some reason a good amount of spam originates from an IP, it may result in it getting blacklisted. However, this has nothing to do with search engines, it has to do with email deliverability.
Normal webmaster behavior would involve linking to third party sites. Say you write an article and you see a great website/page that relates to what it is you're writing about. It makes sense to link to it so that visitors get extra information there.
Chances are that the other website will not be hosted on the same server as your websites though, so its IP will be different. Things will not be that way if you own this other site though, if you are using the same host/IP to host it.
Now, as long as this happens on a small scale, it's not a problem. It can fall under what normal behavior and normal editing processes are about. You may for example refer to your prior work to make a point in a new article. It's to be expected that something like this will occur.
However, if you refer to half of all your prior work to make a point in each of your articles (or on each of your website pages), things go well outside what can be considered normal. Search engines can detect that sort of abnormality, and put a weight on it.
To try and trick the search engine, you can host your website on different IPs. This is a step forward when it comes to making it seem like the websites are unrelated (management wise), however, when your web hosting provider gives you additional IPs, they may be grouped together really close like say x.y.z.122, x.y.z.123, x.y.z.124 etc.
SEO professionals don't want that, because it's not really what you'd expect from sites hosted by different hosts. They want different looking IPs, often referred to as different C-class, meaning that at least the "z" in the IP should change (if not x and y as well). While IP classes have been obsolete since 1993, the term got revived by SEO professionals, and SEO hosting has come to be synonymous to different C class IP hosting. Using more appropriate terminology though, you would be looking for different /24 block IPs.
The IPs themselves are assigned to organizations, and there is a whois system for IPs as well. If all the IPs are assigned to the same entity, it may be enough for the host to question this oddity. Quite a few SEO hosting services won't be able to properly protect you from this.
Can this use of disparate/separate/dissimilar IPs really do the job and hide the real connection between the linked sites? In theory, yes. In practice, the effectiveness will probably be limited. The unnatural interlinking pattern that basically turns your websites into a web ring, can be algorithmically detected, and will be enough to raise a search engine's suspicion. The more websites in the ring, the more the ability to distort search engine results ranking, and the more likely the search engine is to take drastic action.
It makes perfect sense that the number of websites involved in this sort of scheme must play a role in determining what constitutes black hat SEO, and Matt Cutts, Google's official voice, confirms this to be the case in his video blog.
Can SEO/multiple c-class IP hosting help?
To a degree, I believe it can. We need to keep in mind that search engines must be all about probabilities, about giving a score for all sorts of different factors. In this way, SEO hosting can probably make the difference between having your websites penalized for using black hat link building techniques, to not having them penalized, but only in a more or less narrow range of involved websites.
For the average webmaster with 10-20 websites built over a few years time, relatively heavy interlinking of theme related websites should not be a problem. Once you go to hundreds of websites, not to mention thousands of them, I doubt that SEO hosting will make much of a difference. You will actually have to come up with linking techniques that mimic, quite close to perfection, the way a normal article author would link to other resources.
Link to your other sites/pages only when it makes sense to link to them. If you have a page about tomatoes, it probably doesn't make sense, to link to your page about the birth of galaxies.
Search engine optimization is all about helping the search engine find the right content to serve its visitors with the information they are after, and, only indirectly, helping yourself. If you fail to take this approach, sooner or later, the always evolving search engine algorithms will catch up with you anyway.