A measure represents a data column which contains quantifiable data, usually numeric, that can be aggregated. A measure is generally mapped to a column in a fact table.
In SSAS measures are grouped by underlying fast tables, which called Measure Groups. Measure Groups are used to associate dimensions with measures.
Measure are selected while creating Cube, which can later be modified for various usage like its formatting, aggregate behaviour etc. Click here to see how to modify measure for formatting.
Attribute represent one or more column in dimension table. Attribute is used to particular expect of the dimension. For example product colour represent colour attribute of the product dimension, which can later be used for slicing/dicing/selecting the cube values. Each dimension in cube contains one Key attribute, which is generally primary key in dimension table.
Click here to see how to define/update attribute for dimension.
As name suggest Hierarchies is used to define hierarchy in dimension. For example Product and Product Category, National and International region etc. Hierarchy can be defined in Hierarchies table in cube design.
Click here to see how to add/delete user-defined Hierarchy.