It is important to distinguish between an extension that necessitates a detailed planning application and one that falls under Permitted Development (PD). If your planned works fall under the former category you do not need explicit permission from the local authority and a planning application will not be necessary. In this case though, we would recommend that a Certificate of Lawfulness of Proposed Use or Development (CLOPUD) is sought from the local authority in order to create a formal record of the works and their legitimacy.
This post on extension planning permission examines some of the criteria for Permitted Development and, conversely, identifies where planning permission should be sought. There will be a number of seemingly complicated terms in the text to follow but it is actually a pretty straight forward process to follow – just take one example at a time.
Contents:
Article 4 Directions

It is worth noting that PD is generally restricted or removed entirely in conservation areas by Article 4 Directions. The Direction is made by the local authority in order to conserve the special character of the area and will often necessitate a planning application where, otherwise, proposals fall under PD. If you are in a Conservation Area it is worth downloading the CA Design Guidelines for your particular CA (you can find this on your local authority planning portal). This document will generally state whether an Article 4 Direction has been made and which PD rights it restricts or removes altogether.
Another constraint of PD is that, whilst the diagrams to follow illustrate massing that can be achieved without planning, this is on the condition that the proposed extension is constructed from similar materials to the original house so as to blend in. We would normally want to celebrate both the new and the old by creating a clear distinction between the two. As such, if you are looking to create a contemporary extension (as we would) that does not seek to mimic the existing building fabric, then you will need to submit a detailed planning application.
The following is not intended as an exhaustive list of all permitted development rights, rather a focus on those aspects most relevant to central London terraced, semi detached and detached houses. The post explores a number of extensions that would fall under permitted development and, through diagrams, illustrates some of the constraints of permitted development.
Rear Extensions
Extensions to the rear of terraced and semi detached houses will fall under PD so long as they do not extend more than 3 metres beyond the rear wall of the original house. Where the rear wall is stepped each separate step counts as the rear wall.

The above diagrams illustrate plan views. Extension example 1.1 would not fall under PD and would require a planning application because the distance from walls A & B is greater than 3 metres.
The above 3 metre dimension extends to 4 metres in the case of detached houses.
Side Return Extensions
A side wall extension that is not attached to the rear wall but extends beyond the rear wall will be subject to the same restrictions as above.

Height of Extensions under PD
The height of extensions allowed under PD depends on the proximity of the proposed extension to the site’s boundaries. The most relevant constraints are as follows:
Where any part of a proposed extension is within 2 metres of the boundary of the land surrounding the property then the maximum height of the eaves that is allowed for all parts of the proposal is 3 metres. As a result, the majority of terraced houses in central London will be constrained by this 3 metre height rule.
In the event that the proposed extension is beyond 2 metres of the boundary the height restriction is 4 metres.
In order to qualify for a two storey extension under PD, the proposed extension must, in addition to the above 2 metres rule, be at least 7 metres away from the boundary of the land surrounding any house opposite. The diagrams below illustrate this rule.

For the purpose of measuring height, the eaves of a house are the point where the lowest point of a roof slope, or a flat roof, meets the outside wall. The height of the eaves will be measured from the natural ground level at the base of the external wall of the extension to the point where the external wall would meet the upper surface of the roof slope.
For the purpose of measuring height, the eaves of a house are the point where the lowest point of a roof slope, or a flat roof, meets the outside wall. The height of the eaves will be measured from the natural ground level at the base of the external wall of the extension to the point where the external wall would meet the upper surface of the roof slope.
The following example shows the side view of an extension with a pitched roof.

Where there is a flat roof a similar approach should be taken for measuring eaves.

The above is a sample of the more common extensions possible under permitted development. To find out more look at Part 1, Class A of the following document:
Frequently Asked Questions
How long does planning permission last?
Generally you have 3 years from the time of approval to implement the planning permission and begin construction works. Read our full guide on planning permission expiry here.
How long does it take to get planning permission?
Normally around 8 weeks, but it could be longer depending on how large or controversial the plans are. Read our more in depth guide on How Long Does Planning Permission take here.
What is the 45 degree planning rule?
The 45-degree rule is assessed on both plan and elevation. An extension should not exceed a line taken at 45 degrees from the centre of the nearest ground floor window of a habitable room in an adjoining property. Read more on the 45 degree planning rule here.