This is a written undertaking by a third-party to be legally liable to pay you if your customer does not. Since the wealth of individuals is uncheckable, limited company guarantees are preferable. Sole traders and all partners in partnerships are personally liable for their business debts anyway. Directors are not liable for their companies' debts.
Parent companies are not liable for their subsidiaries debts unless they issue guarantees. 'Comfort letters' which confirm the shareholding, are not guarantees and should be rejected.
Banks sometimes guarantee payments to suppliers when the debtor is under-capitalised, e.g. in a start-up, where the bank has lent initial funds and wants the venture to succeed.
Wording of guarantees can be obtained from solicitors, credit insurers and credit management books. Guarantors may insist on limiting the amount and expiry date. Give the wording to your debtor who should ask the guarantor to issue on their own headed paper.
About half of all newly-formed businesses do not last two years! It makes sense to restrict initial credit and, for large orders, to get payment guarantees.