Assuming you have downloaded and installed the software, simply click on the Peerscape icon in the status bar at the bottom of your web browser window.
You will first be asked if you are a new or existing user. Unless you have already created a Peerscape profile on some other machine, click "New User". After entering your name and an optional invitation code (look for it in your invitation e-mail), you will be taken to your Peerscape home page.
In these example screenshots, Barack enters an invitation code from Hillary, and as a result a note about a pending friendship invitation appears at the top of his home page. Clicking through it takes him to Hillary's profile, where he is given the option to accept or decline the invitation. After accepting it, he sees more information in Hillary's profile, including a link to her friends-only wall, on which he writes a greeting.