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Story points are a common name for sizing stories in agile projects. Combined with XpVelocity they provide a technique to aid planning by providing a forecast of when stories can be completed.
When estimating work, a common approach is to estimate in terms staff-hours, such as a programmer saying "this will take me two days to do". Many people in the early days of agile, especially those in the ExtremeProgramming community, found that teams struggled to come up with useful estimates using this approach, even when they applied an approach of IdealTime. We found the most effective way to estimate was to size stories relative to each other, and then use past experience to determine how much could be done in an iteration. 
To determine the points for a story, we compare rough relative sizes. If we are estimating the "fibble the foobar" story, we look for a story of similar size that we've already estimated. We sense it's about the same size as "flipping the synergy bit". Then we look at the story point score for "flipping the synergy bit" and score the "fibble the foobar" the same amount.
A team using story points uses a small range of story points to work with. Common examples might be 1,2,4,8 or 1,2,3,5,8 . Often the top number in the series represents "too big" and should be broken down further. 
Allocating story points should be rapid activity. Discussion should only break out when people have contrasting views on the estimate, in which case its useful to have a discussion as it usually means that something about the story isn't clear. Using a ThrownEstimate is a good technique to move things along quickly.
To form a plan with time, you use XpVelocity.
Some teams don't like using story points, preferring instead to use StoryCounting. I don't have a preference between the two - both seem to work equally well so it's up to the team to try out and go with whichever suits them best.
The ThoughtWorks ebook on estimation provides includes a good Q&A on story points. Kent and I discussed them in more depth in the tasteful green book. Most books that talk about planning and estimation in an agile context discuss story points in more detail.
1: "Story Points" is the most common name that I hear these days, but various terms have been used over the years, often with whimsical names that emphasized their arbitrary nature. I particularly like Joseph Pelrine's gummi bears and Josh Kerievsky's: NUTs (Nebulous Units of Time).
2: This is a Fibonacci sequence
3: Using the top number as too big is saying that a story sized at '8' really means '8 or more'. If you do this beware of using this top number when making forecasts of things like completion time, since '8' can turn into all sorts of numbers when it finally gets broken down. It's usually better to explicitly say its too big to be estimated rather than use a false marker number.
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