"Completing" a map is accomplished by reaching the final red chest on a map. A total of 12 separate maps between the “Adventure Maps” and "Extra Maps" tabs in Adventure Mode must be completed.
Reaching the final chest is accomplished by completing all 80 standard levels you come across in a map with a cumulative total of 156 stars between them. This averages out to about 2 stars per level if you are just doing the base levels.
As you progress through these levels there are other star thresholds that must be met to unlock the next 20 levels and they are as follows:
1 - 20 --- Automatic Unlock
21 - 40 --- 30 Stars
41 - 60 --- 66 Stars
61 - 80 --- 108 Stars
Final Red Chest --- 156 Stars
Each area has one bonus level that is not necessary to continue and is unlocked by 3 starring the mission directly beside it. As these missions are not needed to continue through an area, I would suggest passing them up after unlocking
and getting all 3 stars in an Adventure Map for Fast Levels
The easiest way to keep track of yourself and put in minimum effort in keeping track of your stars after getting these achievements and making sure you are moving on from level to level as quick as possible would be to 3 star maps all the way up to and including level 38 and from then on once you unlock any star requirement in a level you can continue to the next level.
(38 Levels * 3 Stars each) + (42 remaining Levels * 1 Star each) = 156 Stars
Generally speaking, the easiest star that appears is for the "unique" tile that the level contains. such as an optional letter tile (U/K or E/V), Suffixes and Prefixes (EN- or -ION), or Corner letters all being the same letter. These can usually be reached within a few words and therefore much faster than the point requirement star.
If these are not noticeably easy for the level you are on, then generally the other easy star to obtain is for spelling words of a certain length (3 letter words, 4 letter words, etc.). Simply look for the required length words to get the star quickly and move on.Using Hints (Thanks to Tatey for the comment)
If you get stuck on a star or don't want to focus, you can use the "Hint" button (light bulb emblem) to get a free word. Using this highlights a path to take to obtain a word for the farthest left star that you do not have completed. This means that you will generally be getting the hint for the specific length words. If you complete that star then you will be getting 3 letter words until you finish the middle star which is always total points. And if you complete the points star you will get the hint for the final star challenge.