What is database?

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A database is a separate application that stores a collection of data. Each database has one or more distinct API's for creating, accessing, managing,searching, and replication the data it holds.

Other kinds of data stores can be used, such as files on the file system or large hash tables in memory but data fetching and writing would not be so fast and easy with those type of Systems.
So now a days we use relational database management systems (RDBMS) to store and manager huge volume of data. This is called relational database because all the data is stored into different tables and relations are established using primary keys or other keys known as foreign keys.

A Relational DataBase Management System (RDBMS)

is a software that:
Enables you to implement a database with tables, columns, land indexes.
Guarantees the Referential Integrity between rows of various tables.
Updates the indexes automatically.
Interprets an SQL query and combines information from various tables.

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Database: A database is a collection of tables, with related data.
Table: A table is a matrix with data. A table in a database looks like a simple spreadsheet.
Column: One column (data element) contains data of one and the same kind.
Row: A row (= tuple, entry or record) is a group of related data.
Redundancy: Storing data twice, redundantly to make the system faster.
Primary Key: A primary key is unique. A key value can not occur twice in one table. With a key you can find at most one row.
Foreign Key: A foreign key is the linking pin between two tables.
Compound Key: A compound key (composite key) is a key that consists of multiple columns, because one column is not sufficiently unique.
Index: An index in a database resembles an index at the back of a book.
Referential Integrity: Referential Integrity makes sure that a foreign key value always points to an existing row.