CRM Connector for AX is a tool that lets you synchronize data between AX and CRM environments. For example, you can move customers or contacts from AX to CRM and vice versa. Connector uses the web services from AX and CRM to move the data around.
Depending on the error you are getting there are different ways to troubleshoot the issue. If you are getting errors calling an AX web service in AX 2009 go to Basic > Periodic > Application Integration Framework > Exceptions. In this form will be the error you are getting in Connector. Look at the Created date and time and then go to Basic > Periodic > Application Integration Framework > Document history. In this form look for the entries with the same date and time as the exception. For those entries click on the Document logs button and View the XML. Often times Connector will first try to do something like find a customer and if it doesn't find one it will call the create method to create the customer. The document history's xml is the call that was made by Connector into AX or if it is outbound it is the xml document AX sent back to Connector. If you look for documents at the time of your error you will likely get an idea of which record it was trying to process. Once you have the record, start looking at the data in AX to see what is wrong with it. Often times the issue is in an address - likely an imported one.
If the error you are getting is not being thrown by AX or by the CRM service and is instead coming from Connector there is some additional logging you can turn on to troubleshoot. Under C:\Program Files (x86)\Microsoft Dynamics\Connector for Microsoft Dynamics find the Microsoft.Dynamics.Integration.Service.exe.config file. Change the value property from 0 to 4 in the blog that looks like this:
<add name="Microsoft.Dynamics.Integration.TraceLog" value="4" />
Then restart the Connector service. Once the setting is enabled log files will get saved to C:\ProgramData\Microsoft Dynamics\Connector for Microsoft Dynamics\Logs.