When do people wear a kippah?

Orthodox Jewish men wear them all the time. Conservative and Reform Jews might only wear them when praying in synagogue or in the home.

It depends on your level of observance.

The custom of wearing a kippah, or yarmulke, is very old, and it's true origins are lost to time, but today it is generally associated with the idea of reminding the wearer of God's eternal presence, and/or creating a slight separation between the wearer and God.

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The idea of men covering their heads originated in the Middle Ages by French and Spanish Rabbis. The idea then evolved and the popularity fluctuated until the 1800's when the leading recommendation was to always keep the head covered.

A great article outlining the history can be found at the attached link.