Methods for cutting, dividing, and trimming objects

Illustrator provides the following methods to cut, divide, and trim objects:

Divide Objects Below command: Acts as a cookie cutter or stencil, using a selected object to cut through other objects, discarding the original selection. To use this command, choose Object > Path > Divide Objects Below.
Cut Path At Selected Anchor Point button: Cuts a path at the anchor point and the one anchor point becomes two anchor points with one located directly on top of the other. To access this button, select one or more anchor points with the Direct Selection tool and locate the button in the Control panel.
Knife tool: Cuts objects along a freehand path you draw with the tool, dividing objects into their component filled faces (a face is an area undivided by a line segment).
Scissors tool: Splits a path, graphics frame, or empty text frame at an anchor point or along a segment.
Split Into Grid command: Lets you divide one or more objects into multiple rectangular objects arranged in rows and columns. You can precisely change the height, width, and gutter size between rows and columns, and quickly create guides for laying out artwork. To use this command, choose Object > Path > Split Into Grid.
Compound paths and compound shapes: Let you use an object to cut a hole in another object.
Pathfinder effects: Provide various ways to divide and trim overlapping objects.
Clipping masks: Let you use an object to hide portions of other objects.