Any points, lines, or areas that you have drawn on the map consist of one or more vertexes which are the points you clicked when drawing those features. The adjust points tool provides a way to move those vertices.
First click on the feature that you want to move. At the bottom of the feature form you will see an action list. Select the "Adjust Points" action. This will cause all the vertices that make up your feature to be highlighted on the map. Zoom in or out if necessary to see them more clearly. Position your mouse or finger over one of these vertices and press down and drag the vertex to a new location and then let go. Repeat for as many of them as you want. For single point features this is less interesting but it works the same way. Between each vertex you will see a slightly more subtle point. By dragging this mid-point you can insert a new vertex into the polyline or area border. With each new vertex inserted two more midpoints will appear making it possible to change the geometry to any shape you want. When you are finished click the "Done" button at the bottom of the screen. Or click the "Cancel" button to discard your changes.
This option is only available for layers that you are allowed to edit. See Permissions.