What is the easiest free method of encrypting my web traffic? I'd like to be able to log in to sites on my web server without sending my password in plaintext.
Edit: My web server is running on the LAMP stack , although it is a shared host so I don't have root.
https
Use a self-signed certificate.
Tell us your web server software for a detailed implementation description!
As stated above, publishing your own certification is free, however knowing more about your environment, may get you more specific answers. Are you running IIS? What will you be logging into that needs encryption? Are you using Windows Servers on the back end?
Since you don't have root your best bet is to contact your hosting provider and see what they can do for you. You may already have SSL access (try using https://yourdomain.com) using a self-certified key.
You should be able to talk them into installing a StartSSL key for you. This provides you with SSL encryption and browsers won't complain that it isn't signed by a valid Certificate Authority.
use Digest Authentication. Since you're on LAMP, you can configure it on Apache with mod_auth_digest.
Since you are trying to reduce costs, any ssl solutions will probably not be an option.
First it requires a signed certificate that cost a bit, the free ones is not always included in all web browsers.
Second to be able to utilize an ssl certificate your server ip must be dedicated to you. This is not the case in every cheap web hosting option. There are technologies that in the future will make it possible to host multiple ssl enabled sites on a single ip, but it's not here yet.
As mentioned before Digest Authentication is one option that doesn't require ssl certificate or dedicated ip. It's a method of authentication that doesn't reveal your password even though everything else in the communication is unprotected. In Apache this can be applied in individual directories by specific .htaccess files. I'll repeat the previous link on mod_auth_digest. This one is usually already installed on most servers so you won't have to ask you web hosting provider.
You don't always require root access to setup Apache to use SSL, but you will likely need to modify config files, which is either done thru your providers interface, or via files via a shell account. Either way you will need a server certificate; either self-signed, from a major company like Verisign, or one of the smaller free places like cacert.org. As noted by others, this does require a dedicated IP to your server or instance on the server.
I would recommend SSL first, but mod_auth_digest isn't a bad backup idea.