I am a little bit confused between depth of field (DoF) and focus points. Which of them is responsible for making part of the image sharp?
Both play a part in image sharpness.
The focus point is what tells the camera where you want the focus to be. This is where your image will be it's sharpest.
The DOF will determine the range (min and max distance from the camera) in which things will be in focus. The more shallow the depth of field, the smaller the range (distance from the camera also impacts this).
For example, portraits are typically taken with a shallow DOF (large aperture) so only the subject is in focus and background gets a nice blurry effect (bokeh).
On the flipside, landscapes will typically be taken with a deep DOF (small aperture) so that objects in the foreground and background are all in focus.
You can try this yourself. (The faster the lens i.e. the lower the f stop number, the better).
You can try the same experiment with changing the distance between the camera and the subject as well to see the impact.
Of course, you can also do the same experiment and actually take pictures instead, but in that case, choose the same focus point all the time so you will see the difference in how much is in focus as you change the f stop.
Your question appears to reflect the common misconception that everything within the Depth of Field (DoF) is equally in focus. This is not the case at all.
There is only one plane that is in focus for any position of your lens' focus mechanism. If you are using auto focus, then which AF point(s) you have selected will determine on what point the camera attempts to focus. Keep in mind that with many modern AF systems the areas of sensitivity for a given point are much larger than the size of the square/rectangle/dot that you see for it in the viewfinder. The camera will focus on the area of greatest contrast within the area of sensitivity, whether it is directly under the indicator in the viewfinder or not.
In a way, depth-of-field is an illusion. There is only one plane of focus. Everything in front of or behind the point of focus is out of focus to one degree or another. What we call DoF is the area where things look, to our eyes, like they are in focus. This is based on the ability of the human eye to resolve certain minute differences at a particular distance. If the slightly out-of-focus blur is smaller than our eye's capability to resolve the detail then it appears to be in focus. When you magnify a portion of an image by making it larger or moving closer to it you allow your eye to see details that before were too close together to be seen by your eyes as separate pieces of the image.
Since things are gradually blurrier the further they are from the point of focus, as you gradually magnify the image the perceived depth of field gets narrower as the near and far points where your eyes can resolve fine details moves closer to the focal plane. That is why DoF calculations must include the display size, viewing distance, and acuity of the viewer's vision in order to give a meaningful result.