You can build certain structures without having planning permission if the structure comes under the Government's "permitted development rights".
A conservatory will often come under permitted development (if you don’t live in a listed building or in a conservation area) but it will have to comply with a specific set of conditions that govern things like the area, height and width of the proposed structure and how it’s separated from your home.
Read this Government page for full details (including diagrams) or speak to the planning department at your local council to find out if you do or don't need planning permission.
Listed building conservatories
You’ll need listed building consent from your local council planning department, so it’s best to speak to them. Historic England have advice on work to listed buildings in England, as a starting point.
Conservatories in conservation areas
You can build a conservatory in a conservation area under certain conditions. Contact your council’s planning department for advice. Again, Historic England is a good place to start for information on historic buildings.