Some great explanations of what link anchor text is - In terms of optimizing your SEO efforts, use keywords in your link anchor texts that most closely match your URL, title tag, H1 tags and content of your landing page.
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It is sometimes called your link "title". If you plan on doing SEO for any websites, when you try to get links from webmasters or sign up for any directory it is quite common that they ask for your link title... that's your anchor text.
For SEO, it helps search engines associate your link's URL to a specific keyword. Which is why "click here" sort of text, when you are actively working for SEO, should be avoided and only keyword rich text should be used whenever possible.
For forum posting purposes, using anchor text looks a lot tidier than your naked url whenever you are posting a link, regardless of whether it is your own link or not.