The WordPress team recently introduced version 3.8.0 or their very popular flagship product. I have used it on a few sites and it appears to be stable, but a bugfix update almost always follows the release of a major version. Expect to see that very soon.
If you recently updated to version 3.7.0 or 3.7.1, these versions introduced an automatic update feature. This feature will not automatically update a site to version 3.8.0. Although the process was not clear initially, it has since been clarified. WordPress will not automatically update from one level to another, such as from 3.7x to 3.8x. It should automatically update versions, such as from 3.8.0 to 3.8.1.
The only major visible difference I see with version 3.8.0 is that there are now several color schemes available for the admin area. I personally find the new default color scheme to be incredibly ugly and hard on the eyes. It can be manually changed on the User page for each user account. The Admin Color Scheme named Light is probably the closest to the traditional color scheme. You may have a preference for other color schemes. Eight variation are available.
The release of a new WordPress level usually means that most plugins will be updated to take advantage of new features. Although plugin updates are rarely critical, most plugins should be manually updated whenever a new version of WordPress is released. Plugins do not normally automatically update unless they are designed to do so.
My recommendation is to wait for the release of version 3.8.1. A bugfix and security version almost always follows an update to a new level.
What is the difference between a level and a version?
A new level frequently introduces new features and improves the overall architecture of the WordPress core files. A new version focuses on fixing bugs and patching any security holes that are discovered by the WordPress user community. WordPress has the most pro-active development team of any open source product on the planet. Although frequent updates are common, and sometimes annoying, they do help to keep the WordPress platform rock solid.