In an effort to make sure this information is accessible (read: so that I find it when I forget next time), I've duplicated the content here.
For the "Master", "Web", "Sessions" and "Analytics" databases, select the following permissions:
- db_datareader
- db_datawriter
- public
For the Core database, select the following permissions:
- db_datareader
- db_datawriter
- public
- aspnet_Membership_BasicAccess
- aspnet_Membership_FullAccess
- aspnet_Membership_ReportingAccess
- aspnet_Profile_BasicAccess
- aspnet_Profile_FullAccess
- aspnet_Profile_ReportingAccess
- aspnet_Roles_BasicAccess
- aspnet_Roles_FullAccess
- aspnet_Roles_ReportingAccess
For all databases:
In addition to assigning the proper roles to your user for the Sitecore databases, you also want to allow the user to execute stored procedures. For each database, open the "Properties" window, select "Permissions", select your user and select the "Grant" checkbox for the "Execute" permission.
This is all from the 8.2 Update 2 installation guide, which I can't seem to link to due to authentication requirements on http://dev.sitecore.net, but I think the permissions are the same for previous versions of Sitecore as well.