In SharePoint 2010 I have a list. If you right click on a list item or click on the down arrow you get numerous options.

I wish to add my own option here. Would anyone know how I can do this. I only want to do it for certain types of documents i.e. docx.
Create a SharePoint-Feature in which you define a Custom Action. Such a feature can be installed on the SharePoint farm. Research Custom Actions using google, you will find lots of tutorials and examples.
In your case you should use the
RegistrationTypeof the customaction definition to link the action to a specific content-type. (TopTip: use contenttypes to manage your data, not files types).