Edit Vector Polygon Layer

Make a Layer Editable
  1. Editing Polygons or Lines is similar to editing Point features, except that there will always be "segments" or "arcs" connecting the vertexes involved.
  2. Go to any POLYGON layer that you want to practice editing. Right-click on the Layer in the Layers Panel and select TOGGLE EDITING.
  3. Now the features in your layer will be EDITABLE and each vertex that comprises the edges of the polygon features will be overlayed with the RED X
  1. Now engage the EDIT NODE TOOL
  2. Go to any vertex A, click and hold the mouse, then drag it to a new position B, then let go of the mouse click
  3. Now the Vertex will have moved, and the POLYGON that it is part of will be re-shaped.
Note that by editing one polygon's edge, you have created a GAP, or an OVERLAP, when compared to the boundary of the ADJACENT POLYGON! This "gap" or "sliver" will cause topological errors.

To prevent these gaps, you would have to adjust the adjacent boundary to edge-match each updated vertex position.
Typically this is accomplished by FIRST editing one polygon with the corrected boundary edge. Then, after saving those edits, begin editing the ADJACENT polygons, one at a time, and using the TRACE tool to align the other polygon edges with the SHARED BOUNDARY.
TIP: it is sometimes useful to delete vertices of a polygon, along the new edge to be created, before using the trace tool.
These editing functions take considerable practice to learn, just keep in mind that you can STEP BACKWARD when editing using the CTL-Z option, and you can DISCARD changeds when turning off the TOGGLE EDITING function, if needed.
Move a Polygon Feature
Occasionally it might be useful to move an entire feature to a new position (rather than adjusting one vertex at a time).
  1. First engage the MOVE FEATURE(S) tool
  2. Click on the polygon feature that you want to move, and drag it to a new position, then release the mouse
  3. The entire polygon will be "nudged" to the new position, without having touched the relative positions of any vertices that make up the boundary.
Keep in mind that MOVING FEATURES introduces the same shared boundary problems as editing vertices.
See also: TOPOLOGY in QGIS Documentation.