Hi Tom
It's quite oxymoronic isn't it? "Natural" link building is in itself a strange term, while Google advocates that you don't proactively link build, yet it's algorithm still relies so heavily on it.
In the purest sense, if you are ever proactively looking to build or earn links to your site from another Google is unhappy about it. The reality is though that's completely unrealistic (never mind hard to enforce and hypocritical from them).
When people talk about "natural" link building, they are talking about the kind of link building or earning that leaves you to the lowest possible risk from Google. Many people have their different definitions on the topic, but here's mine:
A "natural" link is one that is:
- Editorially earned
- Contextual within the content it is placed in
- Links to your site using a branded term, raw URL or non-commercial anchor text
To explain these further: By "editorially" earned, I mean that you should provide something, be it an article, resource, video or whatever, that warrants being placed and shared on someone else's website. If your content really jumps out and makes me people go "Oh wow, look at this, I want to share it" or "Hey, that's really relevant to my site and my users", then any link you get as credit for that resource is "editorially" earned. What you'll often find is that it usually means that the site's you get links from are relevant and contextual to your own site - eg if you ran a fishing equipment website, you may get links from local angler clubs. You really want to warrant your link being in there by producing something of excellent value.
By "contextual", I mean that your link should appear within the body of the article or written content of the webpage. If you've "editorially earned" your content, this should usually happen - however what has happened in the past, even on Moz, is that authors or publishers would advocate the use of an author biography that contains a link to their site. In my mind, it is far better to have a link surrounded by the content/video/article/etc itself - as in put the link there while you're talking about it - as I feel Google are blanket devaluing author bio boxes (rightly or wrongly depending how you see it). Rather than leave it to chance, I'd rather get the link in the content of the page itself.
And finally, how the site links to you seems to be a big factor in "natural" link building. I think Google (again, rightly or wrongly) always look for an excuse to hold it against you if they find you have links going to your site that contain anchor text that you might want to rank for. It's quite ridiculous, I know, but in order to keep things "natural", I'd say you should only have links that contain your brand, just your URL, or any miscellaneous anchor text (eg "check out what these guys did at the weekend").
So when we talk about "natural" link building that's what I think people mean. Now, is this the way you want to do your SEO? Is it the best way? Is it the most suitable or usable and will it give the greatest return of investment? That's all up for debate. I wouldn't say I'm either for or against this method - and it really does differ depending on your resources and industry. There are plenty of people there doing "unnatural" link building and getting results and ROI. One thing I would say though is that it's hard to build and keep a brand presence if your site is being penalised, so natural link building does come into its fore there.
I hope this helps - as you can see the question makes for a good debate and I hope my point of view helps explain a few things.