Restrictions to online speech

There are multiple laws and voluntary arrangements that restrict free speech on the internet in the UK.

Publishing and social media

Section 127 of the Communications Act 2003 outlaws the sending of grossly offensive, indecent, obscene, menacing or false messages over a public electronic communications network.
The Malicious Communications Act 1988 is also used by police in arresting people under similar criteria to Section 127.
The Public Order Act 1986 (Section 18?) is used in arrests where messages are believed to be inciting racial hatred.
Libel laws. The most recent update to the UK's libel laws (although not all sections apply outside of England and Wales) is the Defamation Act 2013. Regulations to be attached to Section 5 will affect how websites deal with some forms of user supplied content.
Obscene Publications Act 1959
Sites accused of potentially violating trademarks have been taken down by Trading Standards in the UK.
Extremist materials taken down ("Prevent" strategy)

Child-abuse imagery is illegal to possess in the UK, and the Internet Watch Foundation produces the CAIC list, which is used voluntarily used by most UK ISPs to block access to resources hosted outside of UK jurisdiction.
Copyright, Designs and Patents Act 1988 Section 97A was introduced in 2003 to allow copyright owners to apply for injunctions agains ISPs to force them to block access to websites accused of copyright infringement. It has been used multiple times since 2011 (see Website blocking).
Additional websites are voluntarily blocked by UK ISPs in blocking access to "adult content". In most cases access can be granted by users by registering themselves as being over the age of 18 and paying a small fee with a credit card.
Over 1000 items of "extremist" material hosted outside of the UK have been blocked. ^[1]

Blocking and takedown of materials that may promote self-harm or suicide. (e.g. Suicide (Prevention) Bill)
Blocking and takedown of materials promoting gang violence (e.g. Internet Regulation (Material Inciting Gang Violence) Bill)
During the 2011 riots the Prime Minister was considering a proposal to temporarily block access to social media sites such as Twitter and Facebook.