Dimensions represent categories that provide perspective on your numeric data; for example, product category, date, region, cost center, etc. 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), etc.
Measures represent the numeric values that you are analyzing; for example, sales revenue, salary, number of employees, quantity sold, etc. 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, etc. 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.