If you are interested in requesting a letter of recommendation from me (or anyone else for that matter), here are some thoughts that may help you both get a letter in time and that it's a good one.
Keep in mind that not everyone is the right choice to write a letter for you. The best letters will come from professors that know you, have worked with you, and with whom you've ideally interacted outside the confines of a classroom. Sometimes, your letter-writers may indicate that the letter they would write for you would not be a very strong one because they probably don't know you well enough to do a good job. It means nothing beyond just that: that your interactions with this potential letter-writer have not been extensive enough and that you should probably seek a letter from someone that knows you better. This is a good thing, since you'll get a much better letter from someone that is more familiar with your accomplishments.
Making the Initial Request
The best time to make a request for a letter of recommendation from a professor is in person during their office hours, although sending an email a couple of days in advance letting them know that you'll drop-in to make this request would help as well.
Writing a good letter of recommendation takes a good deal of time, and time is hard to find in a hectic schedule. Make sure to leave plenty of time before any deadlines the letter must meet: Request a letter at least two to four weeks before its deadline, and make sure to let your letter-writer know what the exact deadline is.
Take Care of the Logistics
Make sure you provide your letter-writer all the details that are needed to ensure that the letter gets to the right place in time. You'll need to clearly inform them who the recipient is and what the preferred delivery method and letter format are. If the letter must be mailed to its recipient, make sure to provide your letter-writer with a pre-addressed and stamped envelope.
Be Specific in Your Request
You really don't want a generic letter that might work for any and all situations which you may need a letter of recommendation. What you really want is a targeted letter that is custom-written for a specific opportunity that you are applying for.
You'll want to provide your letter-writer with a detailed description of what you need the letter of recommendation for and any special criteria or concerns that are particularly important. That way the letter can be tightly focused and therefore much more helpful to your purpose.
Prepare, Prepare, Prepare
You really want to make the task easy for your letter-writers, and the best way to do this is to provide them with a lot of information:
Remind your letter-writers of your interactions with them and be specific! Remind them what classes or classes you had with them and during which term. If you are particularly proud of a piece of work in these classes (like a project, for example) remind them of this as well. The more specific details you help your letter-writers recall, the better the letter will be.
Provide in-depth material! Remember that your interaction with a professor is a very narrow slice of your achievements, so you really want to provide a deeper pool of information for your letter-writers to draw on. Make sure that you provide a detailed list of activities and accomplishments since you've last interacted with a letter-writer, and always include an updated resume.
Make a strong case! Since opportunities vary with respect to the characteristics you must fulfill, it's a good idea to make your case to your letter-writers that you're a good fit with these characteristics. Write a short sentence or two for each of the things that are important to the opportunity you're applying for, since chances are your letter writers do not know all the details that you know to make that case as strongly. Your English professor, for example, probably doesn't know about your volunteering at the local food bank every week, which may be quite important if a scholarship includes a public service requirement.
No this isn't the same as writing your own letter of recommendation, which you should avoid doing. This is about making it easy for your letter-writer to do the best job they can possibly do.
Keep It Confidential
The best letters are those that are honest and straightforward, and include a realistic perspective that acknowledges challenges and shortcomings. Therefore, you want your letter-writer to feel comfortable with being honest. To ensure this, understand that letters of recommendation are confidential. If you have a choice between keeping letters confidential or not, always choose to keep them confidential.