For example the dimension Month might have the elements "Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", and "Dec". These elements might be rolled up into the consolidation "All Months", or perhaps Jan, Feb and Mar might roll up into "Q1", which in turn rolls up to "All Months".
Elements can typically be numeric (have numbers stored against them), string (have text stored against them) or consolidated (numeric, but they automatically add up their children and are not inputtable).
Dimensions are put together to create cubes.
In ROLAP and data warehouse systems the term "Dimension" may be used in an entirely different sense, i.e. as defined by a star schema or snowflake schema. In that relational database context, a Dimension is like a master data table related to a fact or transaction table.
For example, in the terminology used by SAP Business Warehouse (SAP BW) "Characteristics" are the equivalent of dimensions in TM1, with the "Key Figures" characteristic being the equivalent of the measures dimension in TM1. In SAP BW the term "Dimensions" applies to collections of Characteristics, for which there is no exact equivalent in TM1.
(Reference: TM1 TurboIntegrator Guide (version 9.1) Chapter 3, Analogous Objects in SAP and TM1.)