Trying to ask a favor or to ask for help from another person is delicate. Showing that you are being mindful of the other's time and energy is important. The phrase "do you mind" is a polite way to ask for small favors like borrowing someone's pencil or getting an extra menu at a restaurant.
For example, look at this conversation in which Nick asks Daisy to take him to the hospital.
Nick：Hi Daisy, how are you doing？
Daisy：Hey Nick, I'm doing well. What about you？
Nick：Actually, I'm not feeling very well. Do you mind taking me to the hospital？
Daisy：Of course！
In this situation, Nick is not feeling well, so he wants to go to the hospital. He could have just asked, "can you take me to the hospital," and conveyed the same message. However, he knew that he would inconvenience Daisy. So, he first explained himself and then used the phrase "do you mind" before asking the favor.
Using the phrase "do you mind" is a simple way to show that you are courteous; hence you are more likely to get a positive response. The phrase shows that you are aware that you may inconvenience someone and are grateful for the help. Importantly, a softer tone of voice is also key to conveying a polite message.
Sentence pattern:
Do you mind (-ing verb form) __________？
Other examples:
Do you mind calling a taxi for me？
Do you mind lending me a pencil？
Do you mind carrying my bag for me？