The definition appears in Article 8, Section 20-815 (Interpretations, Rules of Construction and Definitions) of the Subdivision Regulations:
Plat (or Subdivision Plat) — A complete and exact map representing a Tract of land, showing the boundaries and location of individual Lots, Easements, and Streets which has been approved by the Planning Commission and recorded in the office of the County Register of Deeds. The term includes a Replat.
(To Plat as an action) — To Subdivide a property in accordance with these regulations.
The code also defines two sub-types in the same section:
Plat, Preliminary — A map of a proposed land Subdivision showing the character and proposed layout of the Parcel in sufficient detail to indicate its suitability for the proposed Subdivision.
Plat, Final — A map of a land Subdivision prepared in a form suitable for filing of record with necessary affidavits, Dedications, restrictions, and acceptances, and with complete bearings and dimensions of all lines defining Lots and Blocks, Streets, Alleys, Easements, public areas and other dimensions of land.
In short, a plat in Douglas County is a formally approved and recorded map of a tract of land delineating lots, easements, and streets — and the term also covers replats. The word can also be used as a verb meaning to subdivide property per the county's regulations.