What type of motion is exhibited by our earth?

The two main types of motion exhibited by our Earth are rotation and revolution. The Earth rotates, or spins. The rotation is not random, but to a first approximation always has the same orientation. To describe it we say the earth rotates about its axis. The axis is an imaginary line through the north and south poles, around which the earth spins in a west-to-east direction.
The Earth also orbits the Sun. This motion is Earth's revolution. Strictly speaking, the Earth and Sun both orbit their center of mass, called the barycenter, but because the Sun is much, much more massive than the Earth, the barycenter is close to the Sun, and so it is a close approximation to say the Earth orbits the Sun. The Earth's motion around the sun is in the shape of an ellipse. The Sun is actually at one focus of the ellipse. A consequence of this is that the Earth is sometimes closer to the Sun (this occurs during the Northern Hemisphere's winter) and sometimes farther (the farthest point, or aphelion, occurs during the Northern Hemisphere's summer).
The axis of the Earth's rotation is currently tilted about 23.5 degrees from what would be "vertical" relative to the plane of its motion around the Sun. Thus it is rotating, or spinning, at an angle to its larger orbital motion or revolution. This tilt causes variation in the intensity of solar radiation striking the hemispheres during different parts of the year, giving rise to our seasons.
There are other motions. The entire solar system is orbiting the Milky Way galaxy, and in addition, the direction of the Earth's axis undergoes slow changes called precession and nutation. The elliptical shape of the Earth's orbit changes, too. These motions are involved in the cycles of Ice Ages and interglacial periods in Earth's history.