I would rather have 1 backlink from a relevant site than 10 backlinks from non-relevant sites.

Its not about quantity its about quality. Its all about how google preceives you as an authority in the niche with your niche talking about YOU.

A good way to get quality backlinks is to visit bloggers in your niche (area of expertise) and place good and relevant comments on their blogs and then link back to your site.No spam, and no drive by comments like "nice blog" or similar.

I think if you build many links from sites that are not relevant that it might raise some flags. Especially if you are using some sort of automated script to do this. It also depends on how broad your current site's link portfolio is too. If you have a lot of links from many different sources over a long period of time you can get away with a lot more than a new site.

Also don't pay much attention to the pagerank of a site you are trying to get a link from.
No matter if it is pr0 or pr9.
It's always recommended to get links from high PR sites but PR0 can grow up quickly as well. That will be a good investment.

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The best backlinks is from your niche's authorties web site.but if you can't them,I personal recommend high PR sites,I don't think you will be punished by google from non relevant site's backlinks,I get lots of backlinks from non relevant sites,now I have one PR3 site with some keywords page 1 google position.

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if you links approval in high PR in particular page that you got more benefit ..

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According to Matt Cutts, "a link is a link". That gives me enough reason to believe that it doesn't matter if the link is relevant or not. Of course, it's all just talk until proven otherwise. Personally, I go for relevant links anyway, just to be safe, but I haven't tested to see the effects of going for just ANY link. Perhaps I will try getting ONLY non-relevant links and see how it does. I'll wait on the next domain I work on though.

When you have a high quality contents, you'll get a meaning of backlinks naturally.
Anyhow, we can build backlink artificially with any way from bookmarking submission into inject comments in many blog. As Angela V. Edwards teaching is other effort to create the backlink farm. I don't care it has nofollow or dofollow even as relevant or not.

I worked not just for search engine but for human people too. IMHO.

Straight from the horse's mouth:

Interview with Google's Matt Cutts at Pubcon

Though he said people try to prevent spam with "nofollows", keep in mind that only Google adheres to this policy. As far as I know right now, Yahoo has not adopted that policy, so when Yahoo indexes the link, it gets passed on to other search engines and eventually leaks back to Google as a *do* follow link.

Backlinks from authority sites with a large link profile only if you got a some month/year old domain. If you new, the best ones you should get are the lowbies in my opinion because Google will most likely be suspicious if a 1 week year old domain got a PR 7 backlink from the PR 7's homepage.

I think the older a website is the less likely you are to get penalized for non relevant links.

I don't think directory links are now useless but it is possible that they carry less weight these days. However, if you get enough of them they will still give you some link weight and help you to rank for the specific anchor texts you have used.

You need to have a balanced link profile. If all of your links are from non related sites you'll have a hard time rising in the SERPs. Just a few good quality related links will do you a lot more good than hundreds of non related links.