EditorFor vs DisplayFor is simple. The semantics of the methods is to generate edit/insert and display/read only views (respectively). Use DisplayFor when displaying data (i.e. when you generate divs and spans that contain the model values). Use EditorFor when editing/inserting data (i.e. when you generate input tags inside a form).
The above methods are model-centric. This means that they will take the model metadata into account (for example you could annotate your model class with [UIHintAttribute] or [DisplayAttribute] and this would influence which template gets chosen to generate the UI for the model. They are also usually used for data models (i.e. models that represent rows in a database, etc)
On the other hand Partial is view-centric in that you are mostly concerned with choosing the correct partial view. The view doesn’t necessarily need a model to function correctly. It can just have a common set of markup that gets reused throughout the site. Of course often times you want to affect the behavior of this partial in which case you might want to pass in an appropriate view model.
You did not ask about @Html.Action which also deserves a mention here. You could think of it as a more powerful version of Partial in that it executes a controller child action and then renders a view (which is usually a partial view). This is important because the child action can execute additional business logic that does not belong in a partial view. For example it could represent a shopping cart component. The reason to use it is to avoid performing the shopping cart-related work in every controller in your application.
Ultimately the choice depends on what is it that you are modelling in your application. Also remember that you can mix and match. For example you could have a partial view that calls the EditorFor helper. It really depends on what your application is and how to factor it to encourage maximum code reuse while avoiding repetition.