Walser wrote up these brief descriptions of the early 20th century Berlin he encountered as a Swiss visitor. Apparently at that time, newspapers, at least German newspapers, actually had sections for feuilletons. Some of Walser's writings are character studies, some are of particular places, many are simply a detailing of his own sensations.
I thought I'd find out a bit more about the tradition of these missives.
For anybody still perplexed by what this form is, I liked the Wikipedia suggestion that a modern day English equivalent would be the "Talk of the Town" section of The New Yorker. From that example, you can see that its topics are quite wide ranging, but almost always of the comtemporary moment. As is not always true of The New Yorker, but is definitely true of Walser, they tend to be quite subjective compared to more formal journalism, and show off the writer's wit and stylistic skill.
The first use of the term, though, was not for the literary form itself but for the entire section of the newspaper this kind of thing would be found in. Originally, the "Arts and Style" section of the paper was just at the bottom of the page, often separated from the 'real' news by a bar. It was sometimes known as 'the bottom floor' or 'the basement'. It was the French Journal des débats that first brought out a supplemental section which they called Feuilleton in January, 1800. Feuille means 'leaf', so a feuilleton means 'little leaf', or as we might say, a 'leaflet'.
Uh, no, that was not the way I was pronouncing it in my head.
Aside from 'Talk of the Town', I'm not really sure what our current examples of the feuilleton are. It's possible that you might find a good example in the blogosphere. I'm getting the sense that a great feuilletonist might be in need of a great city to fully display his or her skills. But I'm certainly willing to be proven wrong.
I'm updating this blog to add another pronunciation of 'feuilleton' offered by Google, which was suggested by daveblake in a comment below.