Thousands of years ago, when our ancestors looked up at the night sky, they often saw a long irregular band of light dotted with stars. The Greeks saw this as the milk from the breasts of the goddess Hera, the wife of Zeus, and thus it came to be known as the Milky Way.
This is an artist's rendition of our galaxy. Our galaxy, what we call the Milky Way, is what astronomers refer to as a barred spiral galaxy. The bar is that part in the middle extending outward from the core. Further out, the stars, gas, and dust that comprise the galaxy sweep out in a whirl-pool shape that creates the arms of the galaxy. In between the arms are areas of relatively few stars (though not entirely devoid of them).
Our sun, as indicated in this image, is believed to reside in a section of one of the arms around halfway between the edge of the center and the outer edge of the galaxy. Because we are IN the galaxy, our view looks like this:
This is the view toward the center of our galaxy. If you look carefully, you’ll see that the stars seem to get more and more dense closer to the core of the galaxy. In some places, they’re so dense that you don’t see individual stars, but just a lot of light from the cloud of them.
The dark patches aren’t holes in the galaxy, they’re bands of dust and gas between us and those parts of the galaxy. Pictures make it look like those stars you see in the darker patches are being viewed through the gaps, but, in fact, they’re all on this side of the dust clouds.
This image is a larger panorama of the milky way, with the center of the galaxy to our right, and extending out to the left. On the bottom-left, you can see the Andromeda Galaxy, M31, our nearest large spiral galactic neighbor. If you look just a little bit left of the center and zoom in a little, you can make out the North America Nebula, which is near the tail of the constellation Cygnus.
So, yes, we are inside the Milky Way and we ARE part of it. Every star you can see in the sky is in a milky way. Nearly every astronomical object you can see with the naked eye is also in and part of the Milky Way. The notable exceptions are M31, which you can see if you know when and where to look and have dark enough skies, M13 and Omega Centauri, which are globular clusters that orbit around the Milky Way, and the Large and Small Magellanic clouds (which are only visible from the southern hemisphere), which are small, irregular galaxies orbiting the Milky Way. Everything else you can see without a telescope is part of our galaxy.