The following is a quick note about (cold) emailing professors. You have undoubtedly seen that many professors have notes on their websites that simply say “don’t email me.” I’d like to encourage you to selectively ignore that advice but understand why, when and how.
(updated phrasing 05/05/2021)

Are you ...
a current or previous collaborator? Email me
a member of a historically marginalized group in academia? Email me
simply telling me you applied? Thanks, low probability of a reply
mass mailing every professor? Please no

Success in academia, like many aspects of society, is often a function of who you know and networking. Students of famous professors don’t need to email because they know their applications will get read carefully. Further, they may have had the opportunity to meet potential advisors at conferences and so they are already well connected when applying. If you don’t fit this description, as many of us don’t,

  • How are you supposed to break into the “in-group”?
  • How is someone from a smaller lab, a different discipline, or an underrepresented group supposed to change the field?
They have to email. They have to take the chance of getting no reply to at least put themselves out there. If this is you, please email me and everyone else.
Professors get lots of emails, that’s why you’re reading this. Many of them are generic and have nothing to do with their specific research. Let’s say a professor works 40hrs a week and works with 10 students/collaborators.
  • 40 - 10hrs of collaboration meetings = 30hrs.
  • 3hrs teaching, 6hrs preparing, & one office hour = 20hrs left.
  • Thesis committees, graduate admissions, hiring, rec letters, … = 10hrs.
  • Grant writing and staying current on the literature… = 0hrs.
All of these things actually take much longer than what I’ve listed and are worse around paper and grant deadlines, but we still haven’t budgeted any time to reply to emails from our collaborators or anybody else. This means, a professor likely is reading your email while eating lunch, on the bus on the way home, or at night when they are trying to figure out what they forgot to do today. So, a useful exercise might be to ask yourself:
  1. How much time does it take to reply to the email I’m sending?
  2. Is this email actually a DOS attack? [i.e. did it take me much less time to write than it will for them to reply?]

The following a few broad classes of common emails we receive:
❌Generic “please accept me into your program” email -- DOS attack
  • We will read your application eventually anyway, I promise, I will.
  • We have to look through our budgets, look at existing obligations, etc
  • We have to read through your previous research papers to find a fit
This email is unlikely to get a reply

👍Tailored “please accept me - we do similar work” email -- Potentially balanced
  • We still can’t promise anything, we still have to investigate funding, you still need to apply and we still will go through your application and recommendation letters in detail.
  • You brought up a specific intersection in our research and yours. This requires substantially less homework for us to say something helpful and even if we don’t end up working together now, hopefully, we’ll cross paths and learn from each other in the future.
If there is a specific question, a reply is more likely

✋Very specific technical question
  • Love that you’re in deep in the same area as me, but these often are better answered by one of the other authors. Use your best judgment here.
No, maybe, it's complicated. I think we need to talk a little about how admissions works (YMMV at other institutions).

Every school has an admissions committees that does the initial ranking and sorting of candidates. At LTI/RI this is done by faculty, at UIUC and UWashington, this is done ( at least in part ) by current graduate students. I will read every single application that mentions my name in the statement of purpose. For some faculty that might just be 50-75, for others, it may be several hundred applications. I will also try and read every application that mentions keywords that are relevant to my research -- further increasing my load. This process involves reading your statements, your transcript/resume/CV, your letters of recommendation, and possibly skimming a paper you've written. This takes a long time and honestly, this means that we might not get to spend as much time on your application as you were hoping we would. Here's where your email helps or hurts:

If I get an email that meets the above criteria -- I give the application a closer look. That's it.

It's not nothing, but it might not be what you were hoping for. I do not have the ability to unilaterally admit you to the PhD/MLT program. If we are a really good match I can try and put my thumb on the scale, but once admitted, you're under no obligation to work with me (this is a good thing!). But that means that I can't be the only person who is willing to work with you either. This creates a preference for applications that are "broadly" strong and that several faculty agree would be successful if admitted.
When? After applications have been submitted. Because, again, the main thing you're accomplishing is getting a second look at your application.

Who? People you have an existing relationship with.
  • Did we meet at a conferece/workshop/summer-program/etc? Remind me/us
  • If there's a specific PI you are looking to impress -- your letter writer can also email them/us
  • No existing relationship? See next question about being an outsider
If you're feeling discouraged, see initial list of people who should email me regardless :)
Ask for a meeting? It doesn't hurt to ask, but meetings take even longer than emails so -- see time issues above

Meeting at a conference? Great idea! At a conference our schedule is already cleared for talks and social events so it's /much/ easier to find time to meet with prospective students.
That’s 100% fine, and you should say that. You can indicate where you have tried looking to learn more, and details about your interests. I take no offense to receiving an email, it’s when there’s a request attached to it that things get tricky.
You don’t and that sucks, I’m sorry. Normally you can assume they will try and take one or two per year but funding changes as do other commitments. A professor deciding to have a child, work with a start-up, create a new conference, or any number of other things might eat into their time and so they may not take anybody that year.
While flattering, that’s probably not true. Related to my own multimodal research area I can quickly name a bunch of really awesome other people. Here are a few in no particular order that I wish I had more time to work with:
Henny Admoni (CMU)
Joyce Chai (U Michigan)
Yejin Choi (U Washington)
Mirella Lapata (U Edinburgh)
Devi Parikh (Georgia Tech)
Cynthia Matuszek (U Maryland Baltimore County)
Ellie Pavlick (Brown)
Jacob Andreas (MIT)
Yoav Artzi (Cornell Tech)
Desmond Elliott (U Copenhagen)
Jesse Thomason (U Southern California)
That’s legit, and please let me know. It doesn’t mean I can solve it, but with more information, I can at least try and be helpful.
Hopefully this helps: Graduate Application Support Program for personalized feedback.

Also, ask for examples from other people: e.g. Nelson Liu (UW), Rishi Bommasani (Stanford), or Nathan Schneider's Advice
Email me, let’s see what’s up. :)