What do I do when a death occurs?

This very much depends on where the person has passed away.
If at home and expected, then in most cases the GP will issue a medical certificate for the cause of death.  An appointment can then be made at the local registry office to register the death and obtain a certificate. 
If at home and unexpected then this may need to be reported to a coroner, who may request a post-mortem or inquest, to find out why the person died.
If in hospital then they will take care of issuing the death certificate and advising of next steps. The body will usually be kept in the hospital mortuary and then collected by the funeral director.
If abroad then the death needs to be registered according to the regulations of the country and also with the British Consul.

How do I register a death?

You will need a medical certificate of death or permission from the coroner that you can register the death.
The death needs to be registered in 5 days (excluding saturdays and sundays).
An appointment needs to be made at the local registry office.
You will be issued with a certificate for burial or cremation depending on your choices and this then needs to be given to your funeral director.

This is now £11 in England and Wales.  Each additional copy costs the same amount or a Solicitor can be contacted to obtain certified copies, but there are obviously costs associated with this.