When it comes to gambling, there are two strategies available to a country - regulate it, or attempt to ban it. The United Kingdom chose the regulation route and in 2005 passed the Gambling Act. 

Please note: This article refers to the now defunct UK white list which was replaced by specific UK licenses in 2014. All online gambling sites are now required to hold a license with the UK Gambling Commission to be legally allowed to accept wagers from UK customers.

The Gambling Act 2005 is enforced by the Gambling Commission and seeks to ensure that all forms of gambling are conducted fairly and openly, whilst protecting children and vulnerable people.

When it comes to Online Gambling, the act requires certain criteria to be met before an online gambling product can be advertised or otherwise promoted in the UK. Such criteria includes providing a link to the website gambleaware.co.uk and discouraging under age gambling by displaying the 18+ logo.

They must also be licenced and regulated in one of the following recognised jurisdictions: EEA countries*, Alderney, Antigua & Barbuda, Gibraltar, Isle of Man and Tasmania. Together these jurisdictions make up the UK White List, and you may often hear Online Gambling sites referred to as being "White Listed".

Of course, there is no rule that states that you as the consumer have to play or bet at a site that is White Listed, but doing so may give you extra peace of mind.

* EEA Countries means: Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Republic of Ireland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, The Netherlands (note: not the Netherlands Antilles), Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, UK.

References

For those of you who are interested, the actual Act can be found here: http://www.legislation.gov.uk/ukpga/2005/19

Reference to the advertising of online gambling can be found here: http://www.legislation.gov.uk/ukpga/2005/19/part/16

Section 331: Foreign Gambling

(1) A person commits an offence if he advertises foreign gambling other than a lottery.
(2) In this section “foreign gambling” means—
(a) non-remote gambling which is to take place in a non-EEA State, and
(b) remote gambling none of the arrangements for which are subject to the law about gambling of an EEA State (whether by being regulated, exempted, prohibited or otherwise).
(3) Subsection (2) shall apply to Gibraltar as it applies to EEA States.
(4) The Secretary of State may by regulations provide that a specified country or place is to be treated for the purposes of subsection (2) as if it were an EEA State.

And the following ammendment was made in 2007: http://www.legislation.gov.uk/uksi/2008/2829/regulation/2/made

Amendment to the Gambling Act 2005 (Advertising of Foreign Gambling) Regulations 2007

2.—(1) The Gambling Act 2005 (Advertising of Foreign Gambling) Regulations 2007(1) shall be amended as follows.
(2) For regulation 2 substitute—
“2.—(1) This regulation specifies for the purposes of subsection (4) of section 331 of the Act (prohibition on the advertising of gambling taking place or originating from a non-EEA State(2)the places which are to be treated for the purposes of subsection (2) of that section as if they were an EEA State, but only in so far as that subsection applies to remote gambling(3).
(2) The places referred to in paragraph (1) are—
(a)the Island of Alderney,
(b)Tasmania, and
(c)Antigua and Barbuda.”.