To probate a will is to file it with the probate court. This ensures that property correctly transfers and deadlines are met with the IRS, the State, and the court itself.
Before trying to get all of your family members to name you as executor of their estates, it is important to remember that every state (and court) has different rules, their own procedures, etc. Also, fees generally have to be approved, so don't get the calculator out just yet. Finally, a local probate attorney can be a great help throughout the probate process.
Other Notes:
It is true that not all assets always pass through probate, and there are times when a simple form for a small estate under a certain dollar amount can be filed with the State. These processes also vary from state to state. Finally, just because someone died without a will does not necessarily mean one has avoided the probate court. Each state has their own intestate succession laws that dictate distribution of assets. Regardless, a local probate attorney may save you a great deal of time, money, and many headaches.