Cut a circular selection into a vector in Illustrator

There are several ways this could be done.

First draw a circle above the vector, and select both the circle and vector

Then do one of the following:

• Open the Pathfinder, and hit "Intersect". This is a boolean operation which will phyiscally cut the graphic where the circle overlaps it, and delete the parts outside the circle.
• Use the ShapeBuilder tool, and click on the head, then hold down Alt and click on the other pieces to delete them. Essentially the result is a cut shape.
• Do Object > Clipping Mask > Make. This makes a clipping path, and is non-destrutive. The original graphic is not cut.
• Ensure the circle is filled white with no stroke, then open the Appearance panel, double click on Opacity, and click Make Mask. This creates an opacity mask, which like a clipping mask is also non-destructive.
• Ensure the circle has no fill and stroke. Select the Shaper Tool, and draw a squiggly line over the parts outside the circle. This is also creates a non-destructive edit, called a Shaper Group.

• Draw a circle above the shape.
• Select the circle and the shape below it.
• Choose Object > Clipping Mask > Make from the menu.

You can learn more about Illustrator clipping masks by reviewing the Adobe Illustrator Help Files.