It depends on your school. If your school has a premedical advisor, you should make an appointment to talk to him/her about your school's resources.
If you have a premedical committee, then you usually ask a professor to write a letter, mail it to the premed advisor/committee, and the premed advisor/committee will often write a cover letter or committee letter that will be sent to all the schools you choose to apply to. Often, there is a form (at least I had to sign one) indicating that the letters will remain confidential (it is your choice, but it is usually recommended that applicants waive their right to view their letters).
If you do not have a premed advisor/committee at your school, there are services, such as interfolio, through which you can have your letters collected and delivered to the schools you choose to apply to. I did not use interfolio, but there are plenty of individuals on this forum who have, and can give you advice. However, most med schools require applicants to go through their premed committees/advisors if the school has one/them.
When asking for recommendations, ask early! Also, ask the letter writer if they feel comfortable writing a strong recommendation (luke warm/nonspecific recommendations are not going to help your application). I found that my letter writers appreciated receiving a folder/packet of information from me to assist them: a copy of my unofficial transcript, AMCAS personal statement (or even a rough draft to give them a feel for the way you are presenting yourself and your decision to enter the field of medicine), CV, a stamped envelope addressed to the premed advisor's office, and plan for the upcoming year (ECs, etc). I also included a letter to each letter writer, thanking them ahead of time, giving them a date (include some cushion room in case they wait to the last minute so that even if something comes up in their life, your application will not be delayed)by which I wanted to have my letter submitted, and any other relevant information.
That is a lot to think about, but if you search through threads from the summer, there are a lot of applicants who agonized about profs and other letter writers who took a long time to get around to writing their letters...thus delaying their applications...
Good luck! Do research about the application process before your application year so that you will be prepared for whatever the process brings.