The new releases work on Windows 8, Windows 7, Windows Vista, and Windows Server 2008 and 2012, but not on Windows XP. On most Linux and OS X versions only 64-bit Java Runtime Environment is supported.
The changes in the new feature release are many, and that includes security changes as well.
Among them are client-side TLS 1.2 enabled by default, stronger algorithms for password-based encryption, support for AEAD algorithms, enhanced Support for NSA Suite B Cryptography, SSL/TLS Server Name Indication (SNI) Extension support in JSSE Server, better Support for High Entropy Random Number Generation, and more.
If you want to check which Java version you currently user you can do it here, and then download the newest version here. End users running Java on a desktop should go with JRE, as it covers most of their needs, and contains everything required to run Java applications on their system.
But before doing that, you might want to ask yourself if you really do need it, as unpatched Java vulnerabilities are frequently taken advantage by exploit kits. If you choose to install it, be sure to update it frequently.