Yes, Earth does rotate around itself. This rotation is commonly known as the Earth's rotation or the Earth's spin. It takes approximately 24 hours for the Earth to complete one full rotation on its axis, causing day and night as different parts of the planet are exposed to or facing away from the Sun.
As the Earth rotates, the side that is moving towards space is the one facing away from the Sun. This side is experiencing nighttime. Conversely, the side that is moving away from space is the one facing towards the Sun, experiencing daytime. The rotation of the Earth causes the cycle of day and night as different regions of the planet come into and go out of sunlight.