One Way
When you want your Client application should not wait untill your Web Service completes the execution then you may use "One Way" by defining OneWay=true property of SoapDocumentMethod Attribute.
Have a look at below link
http://msdn.microsoft.com/en-us/library/system.web.services.protocols.soapdocumentmethodattribute.oneway.aspx
In this return type of your WebMethod has to be void .
Two Way
This term can be taken in two ways.
1) Web Service returning something back to client.
http://netpl.blogspot.com/2009/07/aspnet-webservices-two-way-response-and.html
2) Service is calling back Client.
For Traditional Web Service you may refer below link.
Duplex Web Services
For WCF Duplex refer this link.
http://geekswithblogs.net/claeyskurt/archive/2007/09/05/115169.aspx

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