Servers store dates in Coordinated Universal Time (UTC). If you don't specify a time zone for your data, it is assumed to be UTC. Web browsers convert the date to local time. Thus, the time offset changes the date display in popups. For example, if you are located in California during standard daylight time, what you see is 8 hours earlier (UTC-8) than the time in the data. Thus a date of 7/7/2014 12:00 a.m. would be converted to 7/6/2014 5:00 p.m. by the browser.
When you publish your data from ArcMap, you can set the time zone on a layer in Layer Properties (Time tab). Because you are providing the time zone, the server will correctly convert local time to UTC time when storing it on the server. Then, no matter where you are in the world, the browser will display the correct time, for the particular time zone. In your case, it probably doesn't matter, But for example, if you're dealing with earthquake data, a web map would correctly indicate that an earthquake happened at 6am California time, but show 9am in New York.
Hope this helps,
Mike