If there is a will, the person named as the executor of the estate should be the one to apply for a grant of probate.
If no will exists, a close adult relative can apply for a letter of administration, which will grant them the same rights as a grant of probate would. The applicant is known as the administrator of the estate, rather than the executor.
Probate is time- and labour-intensive, so some people ask a professional to apply for probate and settle the estate on their behalf.