The best solution is to run a server side mail list manager that runs as a part of your website, using an online database. This is the optimal way to do it, especially because you’ll have the capability of cross-referencing this list with your online contributions database.
My favorite in this group is the PHPList mailer program, which is fairly easy to install and configure. More challenging to install is Dada Mail which runs via the perl scripting language.
Most CMS engines have modules for sending email to a list. Drupal has several, and I’ve used the simplenews module which is convenient, but extremely limited. There’s also a massmailer module which uses the phplist engine to send mail. The advantages of having your list manager as a module in the CMS is that your subscribe pages look exactly like the rest of your website without any extra tweaking.
Above all, if you plan on sending your own email broadcasts, you should know what your hosting provider’s email usage parameters are. N.B. Even though some hosting companies have stated limits to the amount of mail you can send, this isn’t a guarantee—what happens is that when you run your broadcast software it hogs the server’s resources. Most companies will have a background process running, sometimes called “Carnivore” which is designed to kill any scripts which use too much of the server’s resources. You may be able to throttle back the rate at which you’re sending emails in order to keep carnivore at bay. I didn’t have much luck in this department and it’s the primary reason why I deployed my own server.
If you’re launching a major campaign and will be using your own dedicated server, there are a few programs which use perl and sendmail to get the job done, like Majordomo and Mailman. These require a unix system administrator to install them.
If you really want to pay money, there are a ton of services which can send thousands of emails for you, at about .02¢ apiece in bulk. Email services will be provided for free with any good turnkey solution, but most will charge up to a couple of cents per email. This can quickly add up.