The other icon that marks an email's importance is just called the importance marker. Unlike the star, the importance marker will be automatically applied whenever an email comes in that Gmail thinks will be important to you. Some of the criteria that Gmail uses in this decision is which emails you open and which you don't, which emails you reply to, and who you regularly email the most.
Click an email's importance marker to turn it on or off.
Manually marking an email as important will help train Gmail to automatically mark important emails in the future.
Manually marking an email as not important will likewise help train Gmail to leave unimportant emails unmarked.