How do I send an email to my contacts?

Short answer: Sending starts from the Contacts page, not from Promotions: you pick the recipients there, then compose the email in the “Send Email” window.

Steps

Open the Contacts page of the dashboard. Every send starts there; the Promotions page is for preparing content and reviewing history.
For a single recipient: open their detail sheet and click the small envelope icon next to their email address. Its tooltip reads “Send email”.
For several recipients: tick the contacts you want in the list. An action bar appears at the top; click the “Send Email” button.
In the “Send Email” window, optionally pick an email template you have already written, with “Use a template”, then fill in “Subject” and “Message”.
Your default signature is already switched on: you can pick a different one under “Signature”, or “No signature”. Add “Attachments” if you need them, with “From my computer” or “From visuals”.
Tick “Forward replies to me” to receive your contacts' replies by email, on top of them being recorded on the contact's detail sheet.
Click “Send”, then confirm the number of recipients.

The number of recipients kept can be lower than the number of contacts you ticked: contacts with no email address, and those who have not given marketing consent, are dropped automatically.

Writing to more contacts than the ones on the current page

The contacts list is split into pages, and ticking the header checkbox only takes the contacts on the page you are looking at.

As soon as the whole page is ticked, if your search holds more contacts than that, a “Select all {count} contacts” link appears. It extends the selection to every page at once, and the action bar then shows “(all pages)”.

That is how you write to every contact your search found, without ticking them page by page.

If the button is disabled

The “Send Email” button in the action bar goes inactive in four situations. Hover it: its tooltip names the one affecting you. The detail is in Why is the Send Email button disabled.

Going further

You'll know it worked when: A message confirms the emailing is queued, with the number of recipients kept.