Simply, A database is a collection of Data (Information), where database management system is called database management system (DBMS) and if this system maintain a relation, than it called Relational Database Management System(RDBMS).
Here some terms need to more clear which are as follows:
According to webopedia.com,
"Data refers to distinct pieces of information, usually formatted and stored in a way that is concordant with a specific purpose. Data can exist in various forms: as numbers or text recorded on paper, as bits or bytes stored in electronic memory, or as facts living in a person's mind. Since the advent of computer science in the mid-1900s, however, data most commonly refers to information that is transmitted or stored electronically."
In database, Data is made with character and it makes word, row and finally table.
For specific purpose, collection of one or more than one tables makes a database. According to Wikipedia,
a database is an organized collection of data, generally stored and accessed electronically from a computer system.
A database management system is a collection of programs written to manage a database.
Relational Data Model was developed by Dr. E.F. CODD. He developed the relational data model by taking the concept from Relational Algebra in June - 1970.
Relational Data Model has some 12 Rules which are named after Codd as Codd Rules. According to Codd a package can be called as RDBMS only if it satisfies the Codd Rules.