Probate is the process by which legal title of property is transferred from the decedent's estate to his/her beneficiaries. Since you can't take it with you, the court determines who gets it.
If a person dies with a Will ("testate"), the probate court determines if the Will is valid, hears any objections to the Will, orders that creditors be paid and supervises the process to assure that property remaining is distributed in accordance with the terms and conditions of the Will.
If a person dies without a Will ("intestate"), the probate court appoints a person to receive all claims against the estate, pay creditors and then distribute all remaining property in accordance with the laws of the state. The major difference between dying testate and dying intestate is that an intestate estate is distributed to beneficiaries in accordance with the distribution plan established by state law; a testate estate (after payment of debts, taxes and costs of administration) is distributed in accordance with the instructions provided by the decedent in his/her Will.
The cost of probate is either set by state law or by practice and custom in your community. The typical cost to probate an estate is in the range of 3% to 7% of the total estate value.