I absolutely agree to what @sevensevens says about using your contacts you made through conferences and through your advisor or research group during your PhD. You have a much higher chance of getting a response from those contacts (any kind of response, be it positive or negative). I will talk about the case of when you are applying to public postdoc offers published on newsgroups, research group or university websites, and what @sevensevens calls "cold e-mails".
When I was applying to postdocs (computer science, two in question were in the UK - but I would've likely done same regardless of the country), I did send a couple of e-mails, in one case before, in the other case with the application. I think what @sevensevens says is mostly true; it is not likely to help much. I like to think of it as an equivalent of a follow-up phone call after dropping in your job application: polite, if it's short and timed correctly, but unlikely to make a big difference.
That is also how I would recommend phrasing it. Mine was a two-sentence e-mail, no attachments, along the lines:
I have just finished my PhD on topic X and am currently in a
short-term post-doc doing Y. I am very interested to your open post-doc
offer on topic Z as I think it matches my research interests well, and
have submitted an application for it.
Try and phrase X, Y, Z in a way that shows as much understanding in the postdoc and research group you are applying to, but keep it short.
The other e-mail, which I sent before sending in the application, was due to the fact that the position was not a full-blown postdoc, but rather a short one-year position requiring less qualifications, nonetheless in a topic where I could see an even better match to my research. I sent a similar e-mail, additionally asking if I should still apply due to my (over)qualifications.
The result: I got a response to both e-mails. One of them was a short
"Thank you, I will look over your application.", and the other encouraged me to apply. I ended up being offered the short-term position, while not the other one; but I attribute that to the topic match and my CV as sent with the application and doubt the e-mails made a great difference. And the short-term position developed in a full-blown postdoc which extended beyond the initial year.