A dimensional database is a relational database that uses a dimensional data model to organize data. This model uses fact tables and dimension tables in a star or snowflake schema. So in general we can say  dimensional database is just the optimal type of database for data warehousing(Actually from my viewpoint, this is also the drawback for the Data Warehouse, because all the current RDBMS are all target for transaction system and to track the operational data by nature, it is not designed for data warehouse system, so even data warehouse can build on them, it suffer many limitations).

Overview

Data warehouse databases provide a decision support system (DSS) environment in which you can evaluate the performance of an entire enterprise over time. In the broadest sense, the term data warehouse is used to refer to a database that contains very large stores of historical data. The data is stored as a series of snapshots, in which each record represents data at a specific time. By analyzing these snapshots you can make comparisons between different time periods. You can then use these comparisons to help make important business decisions.

Data warehouse databases are optimized for data retrieval. The duplication or grouping of data, referred to as database de-normalization, increases query performance and is a natural outcome of the dimensional design of the data warehouse. By contrast, traditional online transaction processing (OLTP) databases automate day-to-day transactional operations. OLTP databases are optimized for data storage and strive to eliminate data duplication. Databases that achieve this goal are referred to as normalized databases.

An enterprise data warehouse (EDW) is a data warehouse that services the entire enterprise. An enterprise data warehousing environment can consist of an EDW, an operational data store (ODS), and physical and virtual data marts.

A data warehouse can be implemented in several different ways. You can use a single data management system, such as Informix or SQL Server, for both transaction processing and business analytics. Or, depending on your system workload requirements, you can build a data warehousing environment that is separate from your transactional processing environment.

Data warehouse

DW is kind of database that is optimized for data retrieval to facilitate reporting and analysis. A data warehouse incorporates information about many subject areas, often the entire enterprise. Typically you use a dimensional data model to design a data warehouse. The data is organized into dimension tables and fact tables using star and snowflake schemas. The data is de-normalized to improve query performance. The design of a data warehouse often starts from an analysis of what data already exists and how to collected in such a way that the data can later be used. Instead of loading transactional data directly into a warehouse, the data is often integrated and transformed before it is loaded into the warehouse.

The primary advantage of a data warehouse is that it provides easy access to and analysis of vast stores of information on many subject areas.

Data mart

A database that is oriented towards one or more specific subject areas of a business, such as tracking inventories or transactions, rather than an entire enterprise. A data mart is used by individual departments or groups. Like a data warehouse, you typically use a dimensional data model