A "feed" is subscribed to with a feed reader such as Google Reader, My Yahoo!, etc. This allows you to see when new posts arrive from your favourite blogs. There will be either a link, or (depending on settings) you may be able to read the full post (typically minus any pictures) in your reader.
A "reader" is not a program you install. If you have email at any of the main sites (eg. Gmail, Yahoo!, Hotmail/MSN, etc.) there is probably a page somewhere that you can add feeds to.
There are longer and better explanations but that's it in a nutshell!