You will need planning permission if....
Your House is a listed building.
If the fence or wall, is next to a highway used for vehicles (a footpath is part of the highway) - over 1m height requires permission.
Where it is not next to a highway, any fence or wall over 2m will require permission.
Hedges or trees - to whatever height - do not require planning permission! (Even if they are next to a highway.)
You will need to check that there were or are no restrictions on your property though: such as those that apply to 'open plan' estates.