Unless noted otherwise, my answers apply largely to the Ottoman Empire as it existed in the mid-1500s and early 1600s.

An Eyalet was a primary administrative division of the Ottoman empire, governed by an appointed ruler who had to be nominated by the Grand Vizir and approved by the Sultan. While theoretically serving at the pleasure of the Porte, Pashaliks and Beyerbeyliks had a tendency to become semi-hereditary over time. A Eyalet was the highest level of administrative in the Ottoman Empire during pretty much the entire Early Modern Period, and can be thought of as roughly equivalent to a "province" in area, population, and semantic connotation. Many of the Ottoman Eyalets were based on pre-Ottoman-Conquest terrotorrial division. Theoretically, eyalets could be split, merged, established, and disestablished at the whim of the sultan. In practice, arbitrarily rearranging territorrial political boundries tended to alienate the local elites, so the Sublime Porte did it infrequently except in newly-conquered territories.

A sanjak was a sub-unit of a vilayet, with some notable exception in areas with other types of pre-existing political structures, where sanjaks could frequently exist outside of an Eyalet, like a free city or richsrittershcaft, kinda. Sanjak can be though of as roughly equivalent to a modern county-level government or a Germanic ampter. A sanjak would be smaller than a eyalet or vilayet, and governed by a sanjakbey (literally: sanjak-cheiftan).

Sanjaks might then be subdivided into timars, which would have been roughly equivalent to a European manor with one or more villages supporting it.

The above is largely summarized from Halil Inalcik's excellent and authoritative History of the Ottoman Empire: The Classical Age 1300-1600

The vilyaet is a slightly different type of administrative unit that was created as part of the Tanzimat reforms in the 1860s. Vilayets were generally created from the merger of one or more of the older eyalets, and were governed by a Wali who had a greater capacity of local autonomy, but more supervision from the central govenrment, than the old Pashalik system. Many of the post-1864 vilayets were created from the merger of two or more of the older eyalets, with border "rationalized". I know less about how this system actually worked off the top of my head.