Even after death, your debts will still need to be paid.
An important responsibility of your executor is to make sure all of your debts and liabilities are paid and discharged before your assets are distributed to your beneficiaries.
Debts are usually paid out from your residuary estate (i.e. assets within your estate which have not been specifically gifted to a beneficiary). If, however, there remains insufficient assets in the residuary estate to pay off your debts, any specific gifts you have made in your Will may be liable to meet any remaining debts.
There is an exception to this rule in relation to secured debts, the most common example of this being a mortgage. If, for example, you gift a house with a mortgage attached to it to a beneficiary, then the beneficiary inherits both the house and the debt. It is not paid from your residuary estate. This exception, however, can be specifically avoided in your Will should you wish.