My iMac began to ask for my password each time I log on recently. Why is this happening?

Likely because you made a change somewhere.

1. Click the Apple icon in the top-left corner of the screen and select "System Preferences." ... 
2. Select "Security and Privacy." ... 
3. If needed, toggle over to the "General" tab.
4. Untick the box labeled "Requires Password." ... 
5. Enter your Mac's password in the pop-up window.