When attempting to extract Effort using the API, I don't see an attribute called Effort.
Unlike many other attributes for assets, Effort has to be calculated. Each time a team member records their Effort, a record (entry) is made into the Actuals relation. In order to extract Effort, you find all the "Actuals" entries that correspond to the workitem of focus and sum up their Values attribute.
The following recipe uses the rest-1.v1/Data endpoint to obtain Effort data.
The query.v1 endpoint was introduced in Release 13.2, Summer 2013. Please check Support Center -> About to see if you are on this release or later.
To get a list of Effort recorded against a specific task, use this query:
<Server Base URI>/rest-1.v1/Data/Actual?sel=Date,Value&where=Workitem.ID='Task:1217'
To sum up all of the actuals on tasks in a specific project, use this query:
<Server Base URI>/rest-1.v1/Data/Scope/1042/Workitems:Task.Actuals.Value.@Sum