The stars are given for finding the path(s) that use the least amount of tiles drawn by the player for the game to auto-complete the puzzle. A path auto-completes when there are no tile rectangle/square groups on the path and when the path can only connect to one color.
There are exceptions for the final row of puzzles on each page. The point gauge for the puzzle is a good reference because the answer is not always the highest amount of points you can get.
Single move/Moves: Use number of moves specified in top left corner
Longest path: Have the game auto-complete the number of tiles specified in the top left corner
Golden path: A higher difficulty puzzle
As for the in-puzzle menu: (1) return to the puzzle
(2) reset the puzzle
(3,4) sound/music control
(compass/5) turn replay paths on/off
(6) return to main menu.
- Mirakha - about Symbol Link - Game Challenges