How does oscillator frequency stability affect timing?
The "ppm" spec is usually for "initial accuracy". It means that the parts coming from the production line might vary by that much if the frequency is measured right after they are produced (or maybe after some defined burn-in operating time), at 25 C.

If the temperature varies, the frequency will vary in response, over and above what is specified by the initial accuracy spec.

The frequency will also tend to drift over time, an effect that is often specifed as "aging". This effect is also beyond what's specified for initial accuracy.

For example, the part linked in the other question mentioned in Eugene's comment, FX135A, has an initial accuracy of +/- 20 ppm, but it will drift as much as +/- 3 ppm per year due to aging, and .045 ppm per degree C due to temperature. 


  Also, does that automatically imply that any timing based on it is inaccurate to the same degree?


In crude terrms, yes. But you could imagine testing your device against a more accurate clock and storing calibration data to allow your system to correct for the timing error due to initial accuracy.

Edit

As another question says, some vendors specify the overall accuracy including some temperature and aging effects. Be sure to read your datasheet carefully. If they don't say what effects are included, though, your safest bet is to assume the accuracy spec is for initial accuracy only and that temperature and aging effects will add additional frequency uncertainty.