On Linux-based operating system, the kernel (simply called Linux) is largely separate from the rest of the operating system. While drivers and hardware functionality are controlled by the kernel, applications and services are all separate. That means that you can safely upgrade portions your system and keep the same kernel.
On a dedicated server, you generally want to keep the same kernel whenever possible, even if you upgrade applications. This ensures stability and uninterrupted service to your customers and users. However, when your Linux distribution releases a security update that includes kernel patches, you need to upgrade as soon as possible.
On most Linux distributions, the new kernel version will be installed alongside the old one. This way, if anything should go wrong, you can still go back and boot into the old kernel. When upgrading, you will want to make sure you are installing the correct kernel (i.e. i386, x86_64, PAE, etc.). Match your current kernel’s type with the upgrade. Most package management systems will perform the matching automatically.
Once you have updated a kernel, you will need to reboot the entire system. For that reason, it is best to upgrade during low traffic hours, but even if you upgrade in the middle of the day, you can hold off rebooting until later, without any adverse effects on your currently running software.
Finally, it is important to keep in mind that, if you have any custom-compiled kernel modules, such as those required for virtualization, you will need to recompile those modules after the upgrade. Some applications, especially those that automatically compiled the modules as part of the package install process, will automatically regenerate the modules to work with the new kernel.