Windows 7 doesn't support Secure Boot, so if you want to replace Windows 8 with Windows 7, you must disable Secure Boot. Windows 8 does support Secure Boot, and if it came pre-installed on your computer, chances are the computer is configured to use it by default.
In theory, Ubuntu 12.04.2, 12.10, and 13.04 all support Secure Boot. In practice, there seem to be a lot of problem reports about this support, though, so many users end up disabling Secure Boot. My suspicion is that the problems are mostly related to the fact that Ubuntu is still using the old shim 0.1 rather than the newer shim 0.2, but I'm not positive of that. It is possible to install shim 0.2 on any EFI Linux system; see my Secure Boot Web page for details.
Secure Boot does provide security benefits, in that it makes it harder to successfully install a certain class of malware known as a boot kit. This type of malware runs before the OS and so can hide itself in ways that are impossible to detect once the OS is running. That said, disabling Secure Boot won't make the computer any less secure than an older computer would be. Also, most malware is targeted at Windows, so if you rarely boot Windows and do most of your work in Linux, you're less likely to pick up or be affected by malware. (That's not to say that Linux is safe in any absolute sense. Linux has its own security issues, but they tend to be different from Windows security issues.)
Unfortunately, the details of how to disable and otherwise manipulate Secure Boot vary from one computer to another, although Microsoft requires that disabling Secure Boot be possible on any x86-64 computer that bears a Windows 8 logo. Windows 8 should continue to boot after Secure Boot is disabled.