Question: I've asked a professor who I did research with over the summer to write me a letter a recommendation for graduate programs, but he hasn't responded for over a week. I've resent him an email and have received no answer, but a peer who also worked in his lab over summer got an almost immediate response when asking for a letter. We have both done the same research in his lab and known him for the same amount of time. I don't mind if he declines, but I haven't received any word from him. Should I just move on and ask another professor?

Answer: 
Stop by his office (if in town) or pick up the phone. Just ask him directly if he'll do it or not. A lot of people are not so email/text centric. He'll either say yes or no, but at least you'll know the answer. (And if he says no, just drop it and move on--don't argue.)

Personally, I wouldn't cross off and move on, just because of not responding to email. Especially if he's the best one. He is probably just busy. Or a procrastinator.

Voice or face to face are direct. An email is like a letter. Easy to ignore, to get to it later (which becomes never), etc.

When you do confront him if he says, he's unsure of what you did, offer to give him some bullet points and a BRIEF explanation of what you're going after and key features to emphasize. (In the nature of making his job easier.)
No, contact him again. You gain nothing by "moving on", and you deserve a response.

But let me paint a picture for you.

I'm that professor. I'm very busy and get lots of enquires, both IRL and virtual. I try to answer mails as they arrive, but can't always do so. Busy. Interruptions. Classes. Exams to write. Exams and homework to grade. Lost in thought for a day over a nasty research issue. Busy office hours. So, some thing slip. And the new mails keep coming in - piling up, 30 or more per day. And I don't really have the time or mental energy to scan older mails. So, there are gaps. Or maybe I get a chance to scan it, but your email header doesn't wake me up enough to deal with it.

I also don't think to check my junk box in case a stray word in a valuable mail has triggered the mail bot.

Well, I try to get it right, but I'm not perfect and don't have an assistant to remind me about stuff. Also I'm old and so my mind wanders and my kids have issues of their own that I need to deal with.

But, I'm not a monster, just a human. If you ping me again I'll probably say to myself "oh crap, I shoulda done this last week". And likely deal with it.

If you pop into my office (lucky to catch me) I'll think the same thing and probably apologize. I'm not avoiding you and my job is to help you, even if I'm unwilling to write that letter for you. But I realize you need a definite answer.

If you don't ping me then maybe (just maybe) I'll remember that I "shoulda done that" and catch up, but maybe not.

Ping me. It's ok. Let's see how we can fix it. I owe you that. It's part of why they pay me for this wonderful position.