Minnesota has an estate tax. The estate tax is a tax on all of the assets of a decedent before they are distributed to beneficiaries. If the estate meets the filing requirements, the estate pays this tax to the IRS and/or the state in which the decedent lived (prorated to any other states where the decedent had property). Many states have a threshold different from the federal level.
Minnesota does not have an inheritance tax. An inheritance tax is a tax on the beneficiaries of an estate (a tax on what you inherit). If you are a beneficiary, you generally do not have to include inheritance on your income tax return. However, you may have to pay income tax if you inherit an IRA/annuity, etc., which includes the decedent's pre-tax dollars. You should be notified by the estate if this is the case. The income tax is owed to the state in which the beneficiary lives (not the state where the decedent lived).
Additional information is available on the IRS's website (see federal Publication 559).