The Google Apps Sync icon in the Windows system tray is showing up as offline
Either your network connection is down, or Outlook might have switched into offline mode, for example, after a password reset.
To take Outlook out of offline mode in 2007:Go to File > Work Offline.
To take Outlook out of offline mode in 2010, you can go to the send/receive ribbon and click on the Work Offline button.
Once you take Outlook out of offline mode, click on send/receive to restart the sync process.
I'm getting sync errors, or synchronization has stopped altogether
If GASMO syncs indefinitely or stops altogether, you can also try stopping the outlook process and restarting the program. To do this press “ctrl” + “shift” + “esc”. This will open the task manager and you can go to processes and end the outlook.exe process. Once this action is completed you can open Outlook again.
Did you select an Unlimited mailbox size limit in Google Apps Sync? If so your local PST file might have reached its maximum size allowed by Outlook (20GB by default), which can cause errors or prevent Google Apps Sync from syncing with Google Apps at all. You'll need to either limit the mailbox size in Google Apps Sync (see Set a local mailbox size) or follow these instructions from Microsoft to change the size limit of your PST file and store more than 20GB locally.