It's mainly members of the Royal family (even really minor ones) who seem to get police escorts but, for some formal occasions, people like the Lord Lieutenant of a county will also receive an escort.
Senior politicians (e.g. the Prime Minister) might be given a police escort on security grounds.
As Roxie indicates, police escorts are also used to prevent disorder in relation to fans (and teams) travelling to or from football matches. Some (alleged) major criminals or terrorists will also be given a police escort when the prison van takes them to and from court, in order to prevent any attempt to free them (or to attack them).
In some countries police escorts seem to be given to people on the smallest of grounds. For example, many years ago, a pupil of mine captained the South Yorkshire Schools' Under-15 football team when they visited Moscow (when it was still in the Soviet Union). Their coach was escorted by a fleet of police cars and motorcycle outriders, with saluting police officers standing at every junction along their route, where the traffic lights had all been switched to green for them ;-)