The Primary book gets selected automatically when you navigate to the Books page. Only one book can be the primary one. The Web App currently enforces that at least one book be marked as Primary, so in order to remove the Primary setting from a book, select a different book to be the primary one. Edit a book that is not marked as Primary and check its Primary flag. After saving the change, the previous primary book's Primary flag will be turned off. Also, its Current flag will be turned off if the book was used in a different club or if it was completed.
The Current Handbook flag is used primarily in the Windows App. Generally, any book the clubber is currently working in or is planning to work in during the year should be marked as Current. In the Windows App, current books will be given as choices above the sections list to select which book to mark sections for. Also, any Current books are shown in the Current Handbook Status report. This is especially relevant for Trek & Journey clubbers, since they work in Bible Readings at the same time as a main handbook. It's also not uncommon for clubbers to work on finishing a book from a previous year, while working on the new one with the group.
When a book is completed, the Web App will look at all the books assigned to the member. If there is one for the same club that is incomplete and marked as Current, the app will ask if you would like to mark it as Primary. If you choose yes, then the Primary flag will be updated. You'll also have the choice of whether or not to keep the completed book as Current or not.
At the beginning of the year, it's common to assign both the entrance booklet and the first handbook to a clubber, then mark them both as Current and the entrance booklet as Primary. (Tip: If you assign the handbook first, then the entrance booklet, you won't need to change any of the Primary or Current checkbox defaults.) Once the entrance booklet is complete, you'll get a confirmation dialog asking if you'd like to make the next book Primary, along with an option of keeping the completed one as Current. You would normally say Yes to changing the Primary without checking the box to keep the entrance booklet Current. You might consider keeping the completed book Current when they complete a handbook in the middle of the year, so that it continues to show up on Reports. It just depends on your needs.
Also, another note: the Club > Year > Handbooks page in the Web App will list any book that has sections completed within the year, no matter what the Primary or Current flags are set to.