The cursor will change to indicate the move mode. Simply hold down the left mouse button and drag the features to the desired new location.
To move only the selected features(s) either horizontally or vertically, hold down either the 'X' or 'Y' keys on the keyboard to restrict the movement to that axis. Holding down both keys will move the feature(s) diagonally.
To undo a feature move, select the feature(s) to move back and then select the Restore Original Shape from the right-click context menu.
Shift / Offset Selected Features
To shift/ offset selected features a particular distance, right click when the feature(s) to move are selected and select the Shift (Offset) Selected Feature(s) option. This will bring up a dialog allowing you to specify a distance to shift the selected features by in both the X and Y direction or some distance along a specified bearing. The bounding coordinates of the feature in the current projection/ coordinate system will be displayed at the bottom of the dialog that appears to help calculate an appropriate shift/ offset distance.
To automatically move selected point features to the nearest point on the selected line feature(s), selected both line and point features, then right-clicking or go to the Digitizer menu and select the option Move Selected Point Features to Selected Lines from Move/ Reshape Features.
This tool will automatically move the point(s) to the nearest part of the selected lines, and also provide a dialog option to insert a vertex in the line at the location of the point.