Could you explain to me, how I can draw a shape like the one in the following image?
How can I cut any part of circle to get such 4 shapes with Adobe Illustrator?
Create the circle shape.
Remove the fill.
Give it a stroke that is as thick as the width you want.
Go to Object → Expand.. in the top horizontal menu.
Create a rectangle that's as thick as you want the gaps to be.
While you still have the rectangle selected, hold Shift and select the circle. Use the alignment options to center it horizontally and vertically relative to the circle shape.
Select only the rectangle. Press CTRL + C to copy the shape.
Press CTRL + Shift + V to paste it in exactly the same place as the first rectangle.
Right click the currently selected shape (rectangle 2) and go to Transform → Rotate... Enter 90° into the dialog box and press enter.
Select both rectangles and go to the pathfinder. Merge them.
Select the newly formed cross and the outlined circle and, using the pathfinder again, select Minus Front.
If you want extremely fast solution, grab Ellipse tool and select options as on the picture:
If you want perfect solution you should use Pathfinder panel cutting the circle with a cross.
I give you another answer, but there are a lot of options.
Create a circle (Elliple tool->Shift+Click+Drag) and make desired stroke (i.e. 23 points)
Next, press Object->Expand
Next grab Type tool and type "+", then press Object->expand and resize and locate the cross along the circle.
Then select all the shapes and press Minus front on pathfinder
You can do it also with the scissors tool. Invoke it from the tools panel or just by pressing 'C'. With it you can click any path or shape and Illustrator will split it in two as you would do with your scissors. One good option to do that would be to draw the circle, divide with the scissors on top of the four anchor points to create the four quarters and displace them a bit far from the center to create the gaps.