You've got a blog, I've got a blog—heck, your cat's probably got a blog. For the ultimate in ease and simplicity, all you need to do is grab a spot on one of the free hosted services and start writing. Of course, the ease of starting up there is offset by the lack of control you have over the look and feel, to say nothing of the functionality of that canned blog. You can always to to the other extreme and install WordPress on your own server. With that, you get flexibility, but you also have to do some of the heavy lifting, especially when it comes to getting all the technical pieces to work and play well together.
FlatPress splits the difference between these two extremes. You host it on your own server, or with your ISP, or whatever your arrangement is. What you don't need to worry about, however, is access to MySQL or some other high-powered database back-end. That's because FlatPress doesn't use a database; it stores all its data in "flat files"—plain old text files that don't require a database engine to use them. Presumably if you're making a zillion posts a day, at some point you're going to take a performance hit for this, but if you're blogging a normal human amount—or a normal kitty amount—you're probably going to be okay.
Since it's an open-source project, you can grab a copy of FlatPress for free. The machine you install it on will need to be running a web server (Apache is great) and have PHP installed as well.