A person's 'story' is an image that they have uploaded for a period of 24 hours. Once the image is uploaded to your story, your Snapchat contacts are permitted to view this as many times as they wish in that 24 hour bracket. You can upload many stories at one time.
The online story viewer, as per the link above, allows you to view your contacts' stories from your laptop or computer browser. Simply log in and all stories and snapchats will be available on your feed.
The full Snapchat "story"
Snapchat is a mobile application that allows users to send each other one-off pictures for a very short period of viewing time before the image disappears. Its novelty lies in the fact that the viewer can not save or keep the photo. If the viewer takes a screenshot, however, the sender will be notified.
Snapchat is available online here.
