The Moon is unquestionably under the gravitational influence of the Earth, and it is plummeting towards us as you read this. Luckily, it is also moving sideways, which causes it to keep missing us. Here's another approach. Imagine firing a very powerful cannon toward a target ten miles away. The cannonball starts by moving up, then over, and finally down onto the target.
Gravity pulls the cannonball down, in a curve called a parabola.
Use more gunpowder, and the cannonball will move faster and go farther. With a big enough cannon, you could send a cannonball beyond the visible horizon. In fact, with enough velocity, our cannonball could go halfway around the world and land in, say, the Arabian Sea. If we get out of the way before it hits us, our cannonball would be in orbit. If you're wondering how a mass actually generates gravity, or what gravity actually is, look into the work of Albert Einstein , namely general relativity .