Telephone numbers in the United Kingdom are as follows:

Prefixes

01 and 02 - Geographic Numbers
Numbers beginning 01 or 02 are normal phone numbers for home and business telephone lines. 
These numbers are always split into two parts:
 The area code comes first, and is linked to a specific part of the country. For example, the 020 area code is for London and the 0121 code is for Birmingham. When calling between two phones that have the same area code, this part is optional. It is sometimes shown inside brackets to make this clear. The area code can be two to five digits long.
 The second part, usually separated from the code by a space, is known as the 'local number' or 'subscriber number'. It is normally 6, 7 or 8 digits long but can sometimes be only 4 or 5 digits long. This is always unique within an area code. For example, only one person in Manchester could have the number 9460018. 
Examples: (020) 7946 0018, (0117) 504 1102, (01632) 402881, (01647) 61321, (015394) 52749, (016977) 3789.

03 - UK-Wide Numbers
Numbers beginning with 03 are for businesses, government and other organisations that need a number not linked to any single part of the country. The law says they must always cost the same to call as normal 01 and 02 phone numbers from fixed telephones and from mobiles and be included in all call package bundles.

05 - Businesses and Internet Phones
 055 numbers are used by big businesses that need a lot of numbers for their own private phone networks
 056 numbers are used for internet phone services and let people call an internet phone from a normal telephone. Internet phones can also use other types of numbers
 0500 numbers are free to call from fixed telephones.

07 - Mobile and Personal Numbers
These numbers are for mobile phones and similar mobile services:
 070 numbers are 'personal numbers' that people and business use to forward calls as they move around between different phones and can be very expensive to call
 076 numbers are for pagers (except for 07624 which is mobile phones)
 All other numbers beginning 07 are for mobile phones.

08 - Special Rate
These are numbers that are charged at a different price to normal phone calls.
 0800 and 0808 numbers are free to call from fixed telephones. Except for a few specific charity and helpline numbers, calls from mobile phones are not free.
 Numbers starting with 084 are revenue-share numbers and impose a Service Charge of 1 to 5 pence per minute on the caller
 Numbers starting with 087 (except for 0870) are revenue-share numbers and impose a Service Charge of 5 to 10 pence per minute on the caller
 The Service Charge is related to the first six digits of the telephone number called
 Landline providers add an Access Charge of 0 to 10 pence per minute when calling 084 and 087 numbers
 Mobile providers add an Access Charge of 20 to 40 pence per minute when calling 08 numbers. The total price paid can be up to 45 pence per minute and 08 numbers are never included within call packages.

09 - Premium Rate
These are more expensive revenue-share numbers with a Service Charge of up to £1.50 per minute. These numbers are used to pay for services, for recorded information, for dating and for voting in competitions. The caller also pays an Access Charge to their own phone network and this varies from network to network.

Specific Numbers
Operator: 100

Police (non-emergency): 101

National Health Service (non-emergency): 111

Current time: 123 (not available from some mobiles)

Emergency services: 999 or 112

Telephone numbers in Overseas Territories
Telephone numbers in British Overseas Territories do not come under the UK telephone numbering plan. These calls are treated as international calls. Below are the access codes for the overseas territories:

North American Numbering Plan
 Anguilla +1-264
 Bermuda +1-441
 British Virgin Islands +1-284
 Cayman Islands +1-345
 Montserrat +1-664
 Turks and Caicos Islands +1-649

Others
 British Indian Ocean Territory +246
 Falkland Islands +500
 Gibraltar +350
 Saint Helena and Tristan da Cunha +290
 Ascension Island +247
 Akrotiri and Dhekelia +357 (Shared with Cyprus)
 Pitcairn Islands +64 (Shared with New Zealand)

Other websites 
List of telephone area codes
 UK Numbering Policy section of Ofcom's website

Telephone numbers