If a measure is the number, a dimension is the slice by: by month, by product, by customer, by branch. The dimensions are the standing lists of business things your numbers can be grouped and filtered by — and they are shared by every part of the analytics. There is one list of products, one of customers, one of accounts, used by Sales, Inventory, Procurement and Finance alike. That sharing is what makes drilling across possible (2.3): the figures agree because the slices are the same slices.
One idea in this section matters more than the rest. Some dimensions keep their history: when a customer's category changes, the analytics do not rewrite the past — sales made while the customer was a wholesaler stay wholesale sales, and only new sales join the new category. History-keeping dimensions carry a version of the record for each period of its life, and every figure is read against the version in force on its own date. Other dimensions simply show today's values, because nothing in their meaning depends on the past. The subsections below say which is which, and why.