Trimming objects in Illustrator got considerably simpler in Illustrator CS3, with the introduction of the Knife tool. This tool also exists in CS4 and CS5, and allows free-hand cutting of objects. Because of where it is, and where a tool with a similar icon exists -- the Slice tool, that's under the "Crop Marks" tool -- a lot of designers don't know it exists, and it simplifies trimming and cutting excess parts off an illustration in Illustrator tremendously.
Create an object in Illustrator. If you're just using this to practice, use the "Rectangle" tool to make a square.
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Click on the toolbar icon -- the toolbar is usually at the left hand side of the screen -- and click your mouse cursor on the "Eraser" tool. This will pull up the other options, the bottom most option is the "Knife" tool. Select the "Knife" tool. This will change your pointer to a knife shape.
Draw the pointer through your object, from one side to the other, holding down the mouse button the entire time. This will create a path through the object. When the path reaches the other side of the object, click the mouse to complete the path.
Click on the "Selection tool," and click on one part of the object you just cut, and then drag it away – it will come away from the other part, and can be deleted normally.
To cut in a straight line, hold down the Alt key (in Windows) or the Option key (in Mac OS) as you drag the knife tool through the object. This locks the path of the knife tool to the grid.