Here is a description of the Role of School Boards provided by the Center for Public Education. That link is provided by the Wisconsin Association of School Boards which also provides other basic resources such as school board powers and duties and a guide for candidates.
School boundaries are different than other local municipal boundaries, so the Madison School District also covers parts of Fitchburg, Middleton, Verona and Sun Prairie as well as Shorewood, Maple Bluff, Westport and Burke. And, parts of the City of Madison go to school in DeForest, McFarland, Middleton-Cross Plains, Sun Prairie, Verona and Waunakee.
BELLEVILLE - Serves Belleville community, and portions of Montrose, Primrose, Exeter, Brooklyn and Oregon.
CAMBRIDGE - Serves Cambridge area, including Utica, Rockdale, Albion; and some areas of Christiana, Pleasant Springs, Deerfield, (and Jefferson Co: Lake Mills, Sumner and Oakland).
DEERFIELD COMMUNITY - Serves Deerfield, as well as some areas of Cottage Grove, Christiana, and Pleasant Springs.
DEFOREST AREA - Serves DeForest and Windsor communities, and portions of Hampden, Leeds, Bristol, Burke, Vienna, and Madison.
MADISON METROPOLITAN - Serves Madison metropolitan area, including Shorewood, Maple Bluff, Westport and Burke; as well as some portions of Fitchburg, Middleton, Verona, and Sun Prairie.
MARSHALL - Serves Marshall community, Medina, and some portions of Sun Prairie.
MCFARLAND - Serves McFarland community and portions of Dunn, Blooming Grove, Cottage Grove, Oregon and Madison.
MIDDLETON - CROSS PLAINS AREA - Serves Middleton and Cross Plains community, and some portions of Springfield and Madison.
MONONA GROVE - Serves Monona and Cottage Grove communities.
MOUNT HOREB AREA - Serves Mt Horeb, Blue Mounds, Vermont, Primrose, and some areas of Cross Plains, Springdale and Moscow (Iowa Co).
OREGON - Serves Oregon community, and portions of Brooklyn, Fitchburg, Dunn, Blooming Grove, Rutland, Montrose and Union.
STOUGHTON - Serves Stoughton community
SUN PRAIRIE - Serves Sun Prairie and portions of Blooming Grove, Bristol, Burke, Cottage Grove, Hampton, Madison, Windsor & York
VERONA AREA - Serves Verona area community, including portions of Fitchburg, Middleton, Madison, Springdale and Montrose.
WAUNAKEE - Serves Waunakee and some portions Vienna, Westport, Springfield, Madison and Middleton.
WISCONSIN HEIGHTS - Serves Mazomanie and Black Earth communities, and portions of Perry, Cross Plains, Rockbury, Vermont, and Arena.
These links to non-public schools in Dane County provided by Madisonwithkids.com