There is a lot of debate about what makes a link unnatural. Let’s talk about a few of the things we know for sure. Don’t shoot the messenger here, I am not saying I agree with all of this, I’m just saying it’s proven to be pretty solid information.
Google is likely to consider a link unnatural if:
Many/all/most of your links come from pages that are nothing but link pages (often called links.htm or resources.htm). For the record, I totally agree with this one.
You have a large number of links but most of them are coming from the same few sites. That stinks of unnatural. If your site is truly popular, you’ll pick up more than a handful of sites linking to you. (I agree)
There is no connection between the two sites and if the anchor text has been clearly optimized, you can bet it was set up intentionally and isn’t natural. There should be some plausible explanation for why the sites are connected. I think this is mostly true but sometimes sites pick up links that you wouldn’t expect. If it’s the exception and not the norm, then I think it should be OK. The problem starts when you have too many links like this.
The site that is linking to you is low quality and doesn’t look and feel like a “regular” website (with contact info, About us etc). I agree that links from low quality links shouldn’t count or help you but I don’t agree that they should hurt you. It’s entirely possible that a low quality site (or many low quality sites) that don’t know better are trying to improve their site by linking to reputable sites so it’s very possible to have many low quality links that you aren’t responsible for.
You’ve “over-optimized” the page and the link, Google will likely see it as unnatural. What do I mean by over-optimized? Let’s say the page Title tag, the page file name (keyword.htm) and the anchor text in the link match exactly, word for word and there are countless other links to the same page with the same anchor text. That wouldn’t happen by accident and is an indicator to Google that you set up that link intentionally.
You built links too quickly (there is no precise timeline but if it were easy to legitimately get thousands of links overnight, we’d be doing it. It isn’t easy and when it happens that fast, it’s usually an intentional manipulation (with the exception being a fabulous video or infographic that went viral and rampant due to media exposure).
You have lots of links from Blog networks or link farms (low quality sites that are nothing but a collection of links).
Linking is a grey area and is fraught with problems, imperfections and inconsistencies. The best advice I can give is to create really high quality content and use social media to promote it and attract natural links.