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Data Mart - A single star schema, i.e. dimensions and a fact table that stores data at the detail level. A Data Mart covers a specific area of the business: billing, inventory, transactions, claims, etc.
Data Modeling - Performing analysis on the business processes and data to discover attributes of, and relationships between, data elements. An Entity Relationship Diagram (ERD) is the implementation of a data model. Another way to think of it is to discover the business rules of the data elements.
Data Warehouse - The data warehouse to me is more of a logical concept. All of your data marts (with, hopefully conformed dimensions) plus any ODS or active staging areas. By active, I mean updated in real-time by operational systems primarily in support of the warehouse. To me the warehouse is all of the systems that are required to answer the end users' decision support questions across an enterprise. This is different than the Enterprise Data Warehouse (EDW) that is the granddaddy data source for all decision support. A data warehouse can reside in a single data or in many databases.
Data Warehouse Appliance - A data warehouse appliance is a server, or set of servers, that provides an optimized environment for data warehousing, large data volumes and analytical queries. An appliance usually provides a black box approach with integrated CPU, memory, storage, OS and internal networking coupled with high speed external networking.
Database - A database is collection of data. More commonly it is an application that stores data as well as the stored data.
Dimension - A textual table in a star schema. The textual data describes a fact. Dimensions are the context of a measurement (fact). The dimensions included in the primary key of a fact table implement the grain of the fact table.
Dimensional Model - Data Modeling with the goal of extracting facts and defining descriptions of those facts from business systems. The relationships between data elements should be known before Dimensional Modeling starts (by previously performing Data Modeling to, at least, the entity level).
DML - Data Manipulation Language. Language used to add (insert), update, or delete records in a database table.