Story counting is a technique for planning and estimation. Similarly to StoryPoints it works with XpVelocity to help you figure out how many stories you can deliver in a fixed period of time. It differs, however, in that you just consider the number of stories per unit of time and (mostly) ignore their relative sizes.
The rationale for story counting comes from experience. I've heard several teams look at their history and find that over the course of the project, their estimates from using story points are no more accurate than if they had simply counted how many stories were in each iteration. Given that, the effort of calculating story points isn't worth doing.
Using story counting does not imply that all the stories are roughly the same size (although some teams do work that way). Stories can still vary in size, but over time the bigger and smaller stories will cancel each other out, hence a simple count ends up the same.
This doesn't mean that you discard all consideration of relative size. Teams usually put enough effort in to ensure that the stories are within an order of magnitude of each other in terms of effort (so if they were given story points, they would be in a 1-8 point range). 
With story counting, you use velocity in much the same way as usual, the only difference is that velocity is just a sum of stories rather than a sum of story points.
One of the benefits of estimating with story points is that it helps identify poorly understood stories. So when using story counting you need to ensure there is some mechanism to spot stories with hidden blobs of complexity.
So far the teams I've come across using story counting are teams that have already been good at using story points, so it may be that story counting is a technique for more advanced teams. I've found teams work well with both story points and story counting and have no preference between them.
The ThoughtWorks ebook on estimation has a couple of articles on using story counting based on experiences with ThoughtWorks projects. Josh Kerievsky has a good explanation of his switch from story points to story counting.
1: There's still a danger that you can get into difficulty by doing all the small stories first, thus getting a false picture of progress. If you're concerned about this you can do a rough sizing of stories (such as "T-shirt sizing" into Small, Medium, Large, and Extra Large). Unlike Story Points, don't worry about the proportions between the sizes, all you need is to look to see if there are imbalances in the distribution of stories over time, such as all the extra-large stories at the end.