A seismograph station can determine its distance from an earthquake's epicenter by .?

P-waves and S-waves are released by an earthquake at the same time and travel out from the epicentre in all directions. However P-waves travel faster than S-waves so the further away from the earthquake's epicentre the greater the lead the P-waves have on the S-waves.

Observation of the seismogram trace at the seismometer station will allow the scientists to find the P-wave and S-wave arrival times and measure the difference between them. As the P-waves are faster they always arrive first (hence their name - primary wave). The greater the difference between the time that the P-waves and S-waves first arrive (i.e. the greater the lead the P-waves have), the further the distance to the epicentre.

They are able to use the time difference along with an estimate of the speed of the P-waves and S-waves to find the distance from the station to the earthquakes epicentre.

For more information on this please see the related question.