How is reputation earned
For a detailed explanation see How does "Reputation" work? but broadly reputation is earned for receiving upvotes on questions/answers and for approved suggested edits.
What's the whole 'k' thing about
k is shorthand for 1000, so a user with 15k reputation has 15,000 reputation (note that rounding can occur)
So my reputation as displayed here is 15.5k, so 15.5*1000=15,500 (although my true reputation is 15,503).
This is necessary to fit the reputation of some users into the space available. Some users have as much as 650,000 reputation and this has to be truncated somehow. The same applies for question/answer score/views and anywhere else where large numbers exist.
In extreme cases "m" will be used for millions, as Shadow Wizard points out this can be seen on the 'all sites' information screen. So far this hasn't been needed for any users reputation, but it won't be long.