Some links can only be obtained through an incredible amount of time, effort and/or money, while others require a few emails back and forth. Finding out which one has the most value is the tough part. One thing to keep in mind - don't determine the link's worth by page rank. The measure is purposefully inflated by Google for its wide use in search engine optimization.
You can determine value of a link by checking the site's cache/index rate and by checking how well it ranks on search results (use SEO digger). If the site you're targeting ranks among the top 10 search results, then the value is high. If the site is located somewhere on the second page, there's still value, but not as much. Sites on the third page have less value.
To determine the terms for which a specific site ranks you can use rank checking tool like SEO Rush.
Once you've decided to get a link from a specific website, determine the best way to approach a one-way request. If the site is a directory, you won't need much (sometimes $10 - $100 for submissions). If the site is a commercial company, you'll likely need a very good piece of content. If the site is an informational/research hub, than content unavailable elsewhere is the way to go.
Website design, usability and content format also play a big role in getting the links through. Appealing sites "warm" people more than ugly ones.
You can shoot for the link in the first email, but it will not likely produce a good rate of return. Building a relationship with several emails, and then asking for a very relevant link is a better strategy.
The web can seem like a desert, with only text and nicknames on the monitor, but behind them are real people, with real lives, feelings and ambitions. By connecting with them, you increase the chances of getting the link. Sometimes by building a good relationship, links will come by themselves without asking, and sometimes by maintaining the relationship you can meet the person in real life. Aloha.
Phrasing the request for a link wrong can ruin all the effort, so think well. There are several ways to do it, depending on the site you're targeting.
The first way is ask directly if you can get a link. It works if the target site links to similar sites in your space, or has plenty of outside links. If the answer is NO, you might offer to buy it. Usually the response will be positive and the price will come down to your bargaining skills.
Second, you can ask the site owner to have a look at your content and give feedback. If he does look and still doesn't give you the link, you might offer to buy it.
How do I find an authority site?
Authority links are a must these days, and you must create some really good content to get links from them. Authority sites work on the principle of Trust rank, where hub pages distribute trust to all the pages they link to, while linked pages continue passing that trust (to a lesser degree) and so on. Authority sites usually rank on top for competitive terms like: "vacation + city," "lawyer + industry" etc.
How to Get a Link from a .Edu or a .Gov
The only way you'll get a link from these sites is content. You can't buy links from them.
Linkbait
Linkbait requires tons of time, research and effort. No one will link to generic news/recycled content. It usually has to be something remarkable, something that people will talk about. In the book Purple Cow, Seth Godin talks about creating ideas that get people talking automatically spreading the idea for you. He talks of what is partially known as viral marketing, where consumers spread the positive message.
Some linkbait ideas:
Niche news (search engine roundtable), with a razor-sharp focus on the market segment (i.e. real estate news for a city/region), and your commentary. It has to be one topic, though.
The old 101s - they still work if the content is really great.
Attacks - Talk trash about someone and people will link to you. But beware: this hurts more than helps, if done in a clumsy way (i.e. talking about someone who the online community admires).
Provide incentives and discounts (collect a bunch of coupons relevant to the industry/community).
Good luck with your search engine optimization efforts.
Search Optimization 
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