"Probate" is a legal process through which assets are passed, typically with at least some court supervision, to a person's heirs. With limited exceptions, Probate is necessary to pass assets which have not otherwise been legally designated to pass to named beneficiaries.
A Probate estate will usually include personal property, such as money in the bank, jewelry or a car. It can also include real property, such as the person's home.
A Will is often used to direct distribution of assets through the Probate process. Where a valid Will exists, the beneficiaries named in the Will are entitled to receive the distributions designated in the Will. Otherwise, the estate will be divided among the person's heirs under California law.
Unfortunately, the Probate process is tedious and time consuming, so having professional assistance (or at least guidance) is practically essential.
"Trust Administration" is the term used to refer to the process of administering a trust. Usually this consists of paying estate debts and distributing assets to the trust's beneficiaries. It is generally considered to be less tedious and less time consuming than formal Probate, though getting professional guidance at the outset is always a good idea.
For more general information on the Probate process, I recommend reviewing the Sacramento County Superior Court's discussion, found at http://www.saccourt.ca.gov/probate/decedent-estate.aspx.
