Required: Illustrator 9+

1. Hat

Start your hat by drawing 2 basic paths and filling them with Linear gradient (red to darker-red).

2. Shadow

Make a path (shadow path) like the image on the left. Select the top part of the hat, Copy & Paste in Front (Ctrl+F). Now, select the copied path and the shadow path, hold down Alt key and select Intersect shape areas in Pathfinder pallate.
Your path should looks like the image on the left. Repeat this step to make another shadow path for the bottom part of the hat. Then fill these 2 shadow paths with Linear gradient (red to darker-red), in your Transparency pallate, select Multiply.

3. Fur

With your Pen Tool, roughly draw a path like the image on the left and fill it with grey color. Go to Effect > Distort & Transform > Roughen

Roughen Options

In your Roughen window, enter the following setting: Size=8%, Detail=95/in, select Corner points. You may click on the Preview check box to preview your result.
Make a copy of the grey path by holding down Alt + drag (you may use Copy & Paste if you like) and fill it with white. Scale it down a bit and bring it up slightly above the grey path. Select the top part of the hat and its shadow, bring them to front by Object > Arrange > Bring to Front (Ctrl+Shift+]).
Use the Ellipse Tool to make the circle and repeat the previous step to make the furry ball.

Final

Your final illustration should looks like the image shown below.

Have Fun

Now, have fun with your Christmas hat!