Although I don't hear it often, there is a misconception dealing with gravity on the Moon. Everyone knows that gravity causes objects to fall to Earth's surface. For example, if I let go of a pen it falls to the floor because there is a force of gravity pulling it to Earth's surface. Technically the pen is also pulling on the Earth and the two actually move toward each other, but let's not complicate things.
So why does Earth have gravity? Anything with mass produces a gravitational field and attracts objects to it. You have mass and therefore you have your own gravity, but it is extremely tiny compared to Earth's gravity so you never have to worry about it. Moving on to the Moon, it has mass, but it is about 1.2% that of Earth's mass. As a result, the Moon has gravity, but it is much smaller than that of Earth's. The acceleration of gravity on the Moon is about 16.5% that of Earth's. That's why, when you watch old videos of the late 1960s and early 1970s manned lunar landings, the astronauts seem to bounce. They are "bouncing" because gravity is much lower for them. To conclude, yes, there is gravity on the Moon.
There's another misconception that there is no gravity in space. There is in fact, gravity in space, but that will be left for another post.