To "resolve" a debt means to pay it off or otherwise deal with it so that the estate no longer owes anything for it. To be completely safe, you should resolve all debts before distributing any assets to heirs (although in practice, you may not need to follow this strictly if the estate is clearly worth more than it owes).
First, note that if a creditor fails to contact the estate within the local state regulatory deadline period (typically 3-9 months after publication of a probate notice), the debt is barred and the estate has no legal obligation to pay the debt. See Statute of Limitations and Claims Deadlines.
Also note that not all apparent debts are valid. You may be contacted by unscrupulous parties seeking to defraud the estate, or even just organizations with flawed record-keeping. You should be careful to make sure the decedent did actually owe the money, and that he or she didn't already pay it.
Eventually, the estate will have to pay all its valid debts ... unless the estate owes more than it is worth, in which case not everyone will get back all the money owed.
It may be necessary to sell certain assets in order to raise cash so that these debts can be paid. Try to avoid selling assets the will specifically bequeaths to certain heirs, and in fact, it makes sense to have a plan for what you are doing with ALL the assets before you start selling off some here and there. Of course, if it's just selling a few shares of stock or other basically liquid assets, it's not a big deal.
Note that you may want to pay special attention to medical debts, since the estate must pay these within one year of death in order to be able to deduct them on the decedent's final income tax return.
Aside from the big debts, there will also almost certainly be some debts it's best just to pay as they come due, such as the electric bill.
See Pay Off Debt for instructions on recording any debt payments within EstateExec.
If the estate owes more than it is worth, it is considered "insolvent".
You should strongly consider seeking legal help to resolve an insolvent estate, since there are a number of legal statutes governing who should be paid and how much. Fortunately, funeral expenses, compensation for your efforts as an executor, and anyone you hire to help resolve the insolvency, are generally given top priority for access to any estate funds (see IRS IRM §188.8.131.52 for US federal guidelines, and note that individual states sometimes vary these priorities).
When an estate is insolvent (or near-insolvent), negotiation for debt forgiveness is very common, since the debt-holders may end up with nothing if they don't agree to a lesser amount (see Debt Forgiveness below).
Be aware that federal tax debts are considered senior to most other debts, and under 31 USC section 3713(b), the executor is personally liable for any unpaid taxes of the decedent up to the amount of any other debts or distributions the executor did pay. See CPA Journal for additional details.
Finally, note that IRS Publication 559 states that the executor of an insolvent estate is personally responsible for any tax liability of the decedent or of the estate if he or she had notice of such tax obligations or failed to exercise due care in determining if such obligations existed before distribution of the estate's assets and before being discharged from duties.
It's very common for some debts to be completely or partially forgiven after death, especially if the creditor believes that the estate may not have enough money to pay all debts, and that if the creditor doesn't agree to forgive some of the debt, the other creditors will be paid first and there may not be enough money left at the end to pay to anything to the creditor at all.
When some or all of a debt is "forgiven", that means that the person to whom the money was owed has agreed to reduce the size of the debt. You should get any such agreements in writing. You should also be aware that the amount forgiven is considered taxable income to the estate, and that big corporations such credit card companies will almost certainly report these amounts to the IRS.
Of course, few creditors are going to volunteer to forgive their amounts; you will need to negotiate with them if you wish to get anything reduced. If you have hired a probate attorney, this may be something you delegate to them, since they're used to it.
See Forgive Debt for instructions on using EstateExec to record any debts forgiven.
Not every financial obligation of the estate is an "official" debt, and there are certain informal obligations that continue to accrue during the settlement process, such as insurance premiums, property maintenance, utility bills, and so forth.
While it may make sense to wait to pay most debts until the executor has a clear picture of overall estate finances, the executor also has an obligation to protect the assets of the estate, and failing to pay some of these bills in a timely manner may result in unwarranted harm to the estate. For example, failing to pay a utility bill may result in frozen pipes bursting, failing to make a mortgage payment may result in foreclosure, failing to pay insurance premiums may be problematic in the case of theft or fire, etc. The executor should use common sense about the overall situation, and within the confines of the law, act in the best interest of the overall estate.