By now, you should be looking at something like the following.

If you added an SSH key during the droplet creation process, you're probably already familiar with SSH - just connect directly to the displayed IP address with the username "root".

Otherwise, check your e-mail. You should have a root password waiting.

Connecting to the Console in the Browser

DigitalOcean droplets offer in-browser access to the console. Just click that "Console Access" button and enter "root" for the username, followed by the password from your e-mail.  You'll be immediately prompted to replace this with a password of your own choosing, and then you should find yourself with a shell prompt.

Connecting via SSH from Windows

First, download a copy of PuTTY. You just need putty.exe from that link, to start.

Next, enter root@[your ip address] in the "Host Name (or IP address)" field and hit the "Open" button.

Connecting via SSH from OS X or Linux

First, get to a terminal. On the Mac, look for the Terminal in your Applications folder. On Linux, run one of xterm, lxterm, Gnome Terminal, etc. In menus, these can usually be found under "Accessories".

Once you're at a prompt, just invoke SSH like so.

...replacing 192.168.0.5 with your droplet's IP address.