It is very important that you notify the Home Office whenever you change address. If there are any issues with your immigration permission, the Home Office will normally write to you. If the Home Office doesn't have your current address, you might not receive important correspondence from them which requires you to take action or to respond promptly.
How you inform the Home Office that you have changed your address depends on whether your immigration permission is in the form of a biometric residence permit (BRP) or is in the form of a vignette (a sticker in your passport that you obtained before travelling to the UK).
You must also report a number of other changes to the Home Office, for example if you change your name, or your gender, or if you obtain a new passport (this is not an exhaustive list).
Full information can be found on the Home Office website.
If you are a migrant living in the UK or your visa application was made in the UK, you can use this form to:
update your home address
update your correspondence address
update your legal representative's address
update your dependant's details
inform us of a change in your legal representation
UKCISA (The UK Council for International Student Affairs) has produced a table listing the various changes of circumstances that you need to notify the Home Office about, and which form you need to use to do this. Click on the title 'Reporting changes of circumstances' on the link.
You may also be required to notify the Home Office of relevant changes to your studies. We have a separate webpage which explains this.