If asking for permission is what you do to ask if someone can allow you to do something, then making a request is what you do to ask if someone can do something for you. You are asking for permission from your friend if you want to borrow his or her pen, but you are making a request if you are asking your friend to help you write something.
You will say something like this:
Could you please help me write this?
Notice how the sentence is similar to what you use for asking for permission, except that instead of may, we used the word could. We can also use the word can to make requests, and both could and can will always be at the start of the sentence.
Here are some other examples:
Could you please tell me what I missed yesterday?
Can you please describe what you see?
Could you pass me the rice, please?
Can you give me more examples, please?
In the same way, your friend can respond to the questions above with "Yes, I could", "Yes, I can", "No, I could not", or "No, I cannot."