I've been using Subversion for a few years and after using SourceSafe, I just love Subversion. Combined with TortoiseSVN, I can't really imagine how it could be any better.
Yet there's a growing number of developers claiming that Subversion has problems and that we should be moving to the new breed of distributed version control systems, such as Git.
How does Git improve upon Subversion?
Answer
Git is not better than Subversion. But is also not worse. It's different.
The key difference is that it is decentralized. Imagine you are a developer on the road, you develop on your laptop and you want to have source control so that you can go back 3 hours.
With Subversion, you have a Problem: The SVN Repository may be in a location you can't reach (in your company, and you don't have internet at the moment), you cannot commit. If you want to make a copy of your code, you have to literally copy/paste it.
With Git, you do not have this problem. Your local copy is a repository, and you can commit to it and get all benefits of source control. When you regain connectivity to the main repository, you can commit against it.
This looks good at first, but just keep in mind the added complexity to this approach.
Git seems to be the "new, shiny, cool" thing. It's by no means bad (there is a reason Linus wrote it for the Linux Kernel development after all), but I feel that many people jump on the "Distributed Source Control" train just because it's new and is written by Linus Torvalds, without actually knowing why/if it's better.
Subversion has Problems, but so does Git, Mercurial, CVS, TFS or whatever.
Edit: So this answer is now a year old and still generates many upvotes, so I thought I'll add some more explanations. In the last year since writing this, Git has gained a lot of momentum and support, particularly since sites like GitHub really took off. I'm using both Git and Subversion nowadays and I'd like to share some personal insight.
First of all, Git can be really confusing at first when working decentralized. What is a remote? and How to properly set up the initial repository? are two questions that come up at the beginning, especially compared to SVN's simple "svnadmin create", Git's "git init" can take the parameters --bare and --shared which seems to be the "proper" way to set up a centralized repository. There are reasons for this, but it adds complexity. The documentation of the "checkout" command is very confusing to people changing over - the "proper" way seems to be "git clone", while "git checkout" seems to switch branches.
Git REALLY shines when you are decentralized. I have a server at home and a Laptop on the road, and SVN simply doesn't work well here. With SVN, I can't have local source control if I'm not connected to the repository (Yes, I know about SVK or about ways to copy the repo). With Git, that's the default mode anyway. It's an extra command though (git commit commits locally, whereas git push origin master pushes the master branch to the remote named "origin").
As said above: Git adds complexity. Two modes of creating repositories, checkout vs. clone, commit vs. push... You have to know which commands work locally and which work with "the server" (I'm assuming most people still like a central "master-repository").
Also, the tooling is still insufficient, at least on Windows. Yes, there is a Visual Studio AddIn, but I still use git bash with msysgit.
SVN has the advantage that it's MUCH simpler to learn: There is your repository, all changes to towards it, if you know how to create, commit and checkout and you're ready to go and can pickup stuff like branching, update etc. later on.
Git has the advantage that it's MUCH better suited if some developers are not always connected to the master repository. Also, it's much faster than SVN. And from what I hear, branching and merging support is a lot better (which is to be expected, as these are the core reasons it was written).
This also explains why it gains so much buzz on the Internet, as Git is perfectly suited for Open Source projects: Just Fork it, commit your changes to your own Fork, and then ask the original project maintainer to pull your changes. With Git, this just works. Really, try it on Github, it's magic.
What I also see are Git-SVN Bridges: The central repository is a Subversion repo, but developers locally work with Git and the bridge then pushes their changes to SVN.
But even with this lengthy addition, I still stand by my core message: Git is not better or worse, it's just different. If you have the need for "Offline Source Control" and the willingness to spend some extra time learning it, it's fantastic. But if you have a strictly centralized Source Control and/or are struggling to introduce Source Control in the first place because your co-workers are not interested, then the simplicity and excellent tooling (at least on Windows) of SVN shine.