Generally when Apple releases a major upgrade to OS X, it quickly follows with an update to tackle some of the immediate and outstanding bugs with the software. The time frame for this update has been between 13 days (for Snow Leopard) to 26 days (for Lion), and while Mountain Lion has gone for about a month so far in its initial release, the first update is now available and addresses pertinent problems with Thunderbolt audio noise and Mail accounts not working, among other issues.
One problem that people have reported with OS X 10.8 is an audio noise issue when using Thunderbolt-based devices. Over time the audio output on affected systems will become scratchy and filled with static, requiring the users to quit and relaunch the audio application, or otherwise reinitialize the audio driver to clear the problem. Additionally, some users have noticed issues with Apple's Mail program when using accounts administered through corporate Active Directory domains. Another issue is that some users with some configurations have noticed poor battery life.
This update contains patches revolving around Mail, QuickTime Player, storage and I/O performance and implementations (including audio), as well as battery power management; however, it is a relatively small patch at less than 100MB, suggesting that the fixes in it are for more pertinent issues. Therefore, even if you have noticed a true bug or two in the OS, this update may not fix the problem. Nevertheless, some of those who have installed the developer previews of this update have said that it fixed their audio issues and improved Mail performance with Active Directory accounts.
Do keep in mind that some apparent bugs like Finder sidebars not working properly may be more configuration-based issues and not necessarily bugs, so this and future updates may not tackle all issues you may be experiencing.
As a bug-fix release, this update is recommended for all OS X users, but be sure to take precautionary steps against the chance of something going awry in the update by backing up your system fully using Time Machine or a system cloning tool.
The update is available through Apple's Software Update service, which in Mountain Lion has been integrated into the Mac App Store, but is still available in the Apple menu as Software Update. If you have system notifications enabled for the App Store, you should get a notice about the update, offering you the option to directly install the update or open the App Store.