Database rights are a very new addition to the IP stable, and they exist only in the EU and a few other countries. Database rights protect the investment made to create a particular collection of information. According to these laws, whoever invests in the creation of a database gets the exclusive right to extract or reuse (make available to others) substantial parts of the database, or repeatedly extract or reuse insubstantial parts of the database.
Getting database rights
So, if you pay someone to collect data and put it into a database, then you own the database rights on that database for the next 15 years (in the EU at least). If you then offer access to the database on a web site, people can query it and use the information they got out of it, but they're not allowed to download the entire database and share it with others. Also, making another web site that forwards queries to yours and returns the results is not allowed.
Other protections for databases
The individual data items in a database are not protected by database rights, but they may be protected by other IP laws.
For instance, if you pay someone to scan a large number of newspaper articles and put them into a database, then you get to own the database rights to that database (because you paid to make it). However, each individual article is also protected by copyright, which is owned by the newspaper. Simple facts cannot be copyrighted however, so e.g. individual measurements in a database of sensor data are not protected.
A database can also be protected by copyright, if the selection and arrangement of the contents makes it a creative work. If you manually select newspaper articles and order them in a particular way so as to tell a story, the resulting database may be eligible for copyright protection, also in places where database rights do not exist. Furthermore, the data structure of a database (e.g. the DDL description of an SQL database structure) may be protected by copyright, just like software is.
Licensing database rights
Permission to extract and reuse substantial parts of a database can be given to others by the owner of the database rights via a license. Starting with version 4.0, the well-known Creative Commons (CC) licenses include a grant of database rights, making them suitable for use with databases. There is also the Open Database License, which predates CC 4.0, and has a more academic origin.
The default database license at the Netherlands eScience Center is the Creative Commons Attribution 4.0 license. Putting this license on your database will simultaneously license both the database rights and the copyright (if any) on the database itself and on its contents all under the same well-known and widely used terms.