Как отправить письмо с андроида?

Как вариант, можно использовать JavaMail API.

Пример отправки письма:

public class SendMailTLS {
    public static void main(String[] args) {
        final String username = "username@gmail.com";
        final String password = "password";
        Properties props = new Properties();
        props.put("mail.smtp.auth", "true");
        props.put("mail.smtp.starttls.enable", "true");
        props.put("mail.smtp.host", "smtp.gmail.com");
        props.put("mail.smtp.port", "587");
        Session session = Session.getInstance(props,
          new javax.mail.Authenticator() {
            protected PasswordAuthentication getPasswordAuthentication() {
                return new PasswordAuthentication(username, password);
            }
          });
        try {
            Message message = new MimeMessage(session);
            message.setFrom(new InternetAddress("from-email@gmail.com"));
            message.setRecipients(Message.RecipientType.TO,
                InternetAddress.parse("to-email@gmail.com"));
            message.setSubject("Testing Subject");
            message.setText("Dear Mail Crawler,"
                + "\n\n No spam to my email, please!");
            Transport.send(message);
            System.out.println("Done");
        } catch (MessagingException e) {
            throw new RuntimeException(e);
        }
    }
}