The usual process is that you actively approach those that you think will write you the best letter. In practice, you will want to engage the advisors that are the best trade-off between the following, sometimes conflicting, criteria:
- Prefer people that know you, your work, and your skills in detail over people that you don't know that well. Specifically prefer people that know you from research projects or other "outside the classroom" activities. Letters that only say "ASimonis01 had an A in my class" are not worth much.
- Prefer people that are willing to write you a very strong letter over people that may write a more lukewarm recommendation. This should be obvious, but keep in mind that everybody does that, so the overall level of enthusiasm in letters is very high. Even a well-reasoned positive-ish letter with some negative aspects may already be a kiss of death for your application.
- Prefer people that are in a position to write a plausible recommendation. In practice, this usually means that more senior and/or more well-known recommenders are better than less experienced ones, simply because a statement like "among the best student I have had" means much more if the writer is a senior professor than an assistant prof. in his first year.
- Prefer people who know how to write a recommendation letter. Especially if you are currently located in a different cultural region, it is well possible that your letter writers are not overly familiar with how a recommendation letter is supposed to be written. The end result may be a letter that is meant to be positive, but which is perceived as lukewarm or unprofessional. In the same context, avoid like the plague people that you suspect may just submit a stock letter.
- Prefer academics over pretty much anybody else. That is, don't get recommendations from your boss in industry, your coworkers, or your childhood friend. The reason is the same as above - you want letters from people that are able to judge whether you will be a good grad student. Your boss in industry can't do that. Researchers in industry research labs may qualify, but I would still prefer a professor in almost all cases.
Also note that at least in the US one generally does not get to see the written letters, they are uploaded by the recommender directly. This means that you need to be very careful in your selection, as you cannot pivot after receiving a suboptimal letter from one of your references.
