1. The executor of the will needs to apply for probate. Unless the estate is unusually complicated it's normally a very straightforward task, simply involving a bit of form-filling. (I managed it easily enough anyway!). It can be done either online or by post (using form PA1P). The staff at your local district probate registry can help you with the form-filling.
2. Where a person has died intestate, someone needs to apply for 'letters of administration'. There's a fixed 'pecking order' as to who has the right to apply, in descending order as follows:
the spouse or civil partner of the deceased ;
a child of the deceased ;
a grandchild of the deceased ;
a parent of the deceased ;
a sibling of the deceased ;
a nephew or niece of the deceased ;
another relative of the deceased.
Note that there's no 'pecking order' within each of the categories above. For example, if the person died without leaving a surviving spouse or civil partner, but with several surviving children, any one of those children can seek letters of administration. (i.e. the oldest child doesn't have 'seniority' over his/her siblings).
An application for letters of administration needs to be made by post (using form PA1A).