As Tom Harrington's answer notes, you can produce a colloquial representation of a moment or a time interval using NSDateComponentsFormatter.
However, if you want to do exactly what the question asks in its example, which is to produce a colloquial representation of a moment in the past, relative to the present moment, like for a timeline-oriented UI, then it seems like NSDateComponentsFormatter is not suitable. As the documentation for stringFromTimeInterval(_:) says, the time interval value "must be a finite number. Negative numbers are treated as positive numbers when creating the string."
As near as I can tell, the best choice is TTTTimeIntervalFormatter, a standalone class in Mattt Thompson's FormatterKit.
I have produced an Xcode 7 playground, RelativeDatePlayground, that compares the outputs of NSDateFormatter output to TTTTimeIntervalFormatter. Here is a table showing output for different relative times in seconds. As you can see, NSDateComponentsFormatter does not seem to handle past moments or the present moment well:
        -1488010 |               2 weeks ago |         -1 week remaining
        -1468800 |               2 weeks ago |         -1 week remaining
         -864000 |                1 week ago |       0 seconds remaining
          -86400 |                 1 day ago |          -1 day remaining
          -36000 |              10 hours ago |       -10 hours remaining
           -3600 |                1 hour ago |         -1 hour remaining
            -600 |            10 minutes ago |     -10 minutes remaining
             -60 |              1 minute ago |       -1 minute remaining
             -10 |            10 seconds ago |     -10 seconds remaining
              -1 |              1 second ago |       -1 second remaining
              -0 |                  just now |       0 seconds remaining
               0 |                  just now |       0 seconds remaining
               1 |         1 second from now |        1 second remaining
              10 |       10 seconds from now |      10 seconds remaining
              60 |         1 minute from now |        1 minute remaining
             600 |       10 minutes from now |      10 minutes remaining
            3600 |           1 hour from now |          1 hour remaining
           36000 |         10 hours from now |        10 hours remaining
           86400 |            1 day from now |           1 day remaining
          864000 |           1 week from now |          1 week remaining
         1468800 |          2 weeks from now |         2 weeks remaining
         1488010 |          2 weeks from now |         2 weeks remaining