The name of the small cap male Jews wear is called a Kipa (dome or covering) or Yarmulke. This head covering was meant to signify recognition that God is present above oneself. It was traditionally worn only during prayer or during religious rituals. Eventually, Jews started wearing the kipa all day long, both as a sign of piety and to distinguish themselves from non-Jews.
Today, the kipa is worn all day long by fully observant Jews, although many Jews wear one during prayer or at funerals even though they don’t wear one all day. Different types of religious Jews wear different types of kipas to publicly demonstrate their unique religious outlook.