How to apply for postdoctoral positions
Applying to Positions in Academia in Canada and the United States
Applying to postdoctoral positions is strange, because there are so many different types of them, and postdocs in general exist in a nebulous cloud within academia. The most important thing is that if a prospective postdoc advisor asks you to include items in your application, then you should include them.
Oftentimes, people find postdoc positions because they send an email to a professor whose work they like, and with whom they'd like to work. This professor may or may not have an actual ad for postdocs. This can be problematic because it creates more barriers for marginalized folks in academia, but that's a discussion for another website.
There are some postdoctoral programs at universities that provide independent funding for your research, where you just apply to the program. But for most postdoctoral positions, you apply to work with a particular professor. In that case, there are some general guidelines to follow. Again, if there is a job ad, make sure to provide everything the ad asks for.
Email
The email is your first point of contact with a prospective advisor, so make sure you spend time on getting this right. Unless the advertisement specifically asks for a cover letter, an email will suffice.
Things to do:
Open your email with Dear Dr. [lastname of person] or Dear Prof. [lastname of person]
Explain why you are interested in being a postdoc in this person's lab. It's a good idea to refer to work that they have done, and why you find it interesting.
Explain what qualifies you for the position. At this point, you have (or will soon have) a PhD, so you have research experience. Make it clear how your research experience makes you a good candidate for this lab. Don't just apply to every postdoc position you see come up - make sure you're actually interested in the research that's happening in the lab, and that you have something to contribute.
Bonus: give specifics about what you would like to study, or project ideas. This is not mandatory for entering a new lab, but will certainly help boost your application.
Make sure to proof-read your email. Give it to someone else to read over. If you type it up, don't send it right away. Sleep on it and read it again the next day before sending it.
Things not to do:
Do not use salutations that imply the professor's gender. In the United States and Canada, the custom is that you refer to professors by Dr. or Prof. Do not use Ma'am, Ma, Mr, Mrs, Ms, or Sir. Also do not refer to the professor by their first name. If they reply to you and sign their email with their first name, then you can use it, but it's still generally good to lean toward the formal rather than the informal.
Do not send the same email to every potential advisor. If you follow the things to do above, you will be writing a unique email to each prospective advisor. Yes, this takes some work, but this potential advisor will be investing a lot of resources (money, time, emotions) in you, so the least you can do is send a custom email.
Try not to take it personally if you don't get a reply, or if you get a rejection. If you don't get a reply, it's ok to send a follow-up email (wait a week or two first), because yours might have been missed. If you get a rejection, it's hard, but it's also just a part of life. The more you do this, the easier the rejections become.
CV
The CV is what shows the prospective advisor the things that you described in your email or cover letter.
Things to do:
Send this as a PDF
Follow these general guidelines
Make sure to proof-read
In addition to publications, if you have manuscripts that are in progress but not yet submitted, include them in a separate section.
Go online and check out CVs from people at a similar career stage as you to see the level of information you should include. Plenty of people have their CVs available online.
Things not to do:
This is not a resumé, so keep in mind that the format should be a bit different. Follow the guidelines in the link above.
Do not include personal information. In some countries, this is the norm, but in Canada and the United States, personal information such as religion, gender, marital status, and country of origin should not be included.
Do not include a photograph. Again, this is a regional cultural thing. In many places around the world photos are often included in applications, but not in North America.
Other pieces
If the posting for the position asks for a cover letter, include that as a separate PDF. Make sure to use an official letterhead from the institution you are at (or that you were most recently at). Follow many of the same guidelines as for the email as described above. If you include a cover letter, then you don't need as much detail in your initial email. Just briefly describe who you are, what you're applying for, and what you have attached in the email.
In your email you may want to include PDFs of your publications if you have any (at most three). This saves the prospective advisor time and effort in having to track them down.
By this point in your career, you should definitely have a Google Scholar account, so link to that. And if you have made a personal website, that's a great bonus. Make sure to link to that too.