Database right is similar to copyright, but distinct. Both database right and copyright may subsist in a single database quite independently. However, this note is concerned exclusively with the protection afforded to websites by database right, not copyright.
In the UK, the key piece of legislation is the Copyright and Rights in Databases Regulations 1997. Many of the questions you may have about database right can be answered by reading the Regulations.
A "database" is defined in the legislation as "a collection of independent works, data or other materials which- (a) are arranged in a systematic or methodical way, and (b) are individually accessible by electronic or other means". In some industries this might be called a "dataset" rather than a database. Nonetheless, it is databases so defined that are the subject of the 1997 Regulations. Database right will subsist in a database if there has been a substantial investment in obtaining, verifying or presenting the contents of the database.
The leading case on the meaning of this is British Horseracing Board v. William Hill. In that case the court ruled that BHB's database of runners and riders fell outside the scope of protection because the resources used to create the database did not constitute an investment in the obtaining and verification or presentation of the contents of the database.
If your website has an integrated database that can overcome the limits on protection set out in the Regulations and in the BHB case, then any person who extracts or re-utilises a substantial part of your database without your consent will likely infringe the database right. For example, data mining and scraping activities often infringe.
As the owner of the database right, you may be entitled to damages and an injunction against any infringer. One general exception to protection is this: individuals who are not EEA nationals, or habitually resident in an EEA state, and companies that are not incorporated in an EEA state, do not usually get the protection of the database right.