Dimensions represent categories that provide perspective on your numeric data; for example, product category, date, region, cost center, and so on. Dimensions can contain properties that further describe a dimension. For example, you may have a dimension for customer which has properties such as phone number and address to further describe the customer dimension.
Dimensions can also be rolled up into a hierarchical view; for example, time (year, quarter, month), geography (country, region, location), employee structure (executive, manager, employee), and so on.
Measures represent the numeric values that you are analyzing; for example, sales revenue, salary, number of employees, quantity sold, and so on. Sometimes these quantities are contained in a single dimension referred to as an Account type dimension (and probably with the name Account, or something similar). In this situation, the numeric values represent the line items on a corporate balance sheet, income statement, profit/loss statement, and so on. But you can also present the numeric values as individual elements called Measures.
Together, dimensions and measures are the framework for viewing data, whether it be a trend line of revenue over time or a tabular comparison of gross margin across different regions.