.deb files are Linux packages. You can't simply install a Linux package on Android: Android is not a Linux, it is a separate operating system that only shares the kernel with Linux.
Since Android uses the same kernel as Linux, you can install a Linux system and run Android programs and Linux programs at the same time. You'll need to install not just the program you want, but all the supporting software, in particular libraries.
Not all Linux software can work on Android. The Android kernel is compiled with different options from a normal Linux kernel, though the missing features¹ aren't used by many applications. Low-level system software (e.g. things that require an init script) are likely to be hard to install as they conflict with Android's equivalents. Anything that requires systemd (e.g. Gnome 3) is out without some serious work that I think hasn't been done. Anything with a GUI requires an X server, but there are X servers for Android, even some on Google Play.
There are several prepackaged ways to install a Linux system on Android. Most of them require a rooted system. Kernel versions since 3.8 (run
uname -r at the adb prompt to check what you have) make it possible to go a long way without rooting, thanks to user namespaces, but I don't know if there are installers that use this feature. If you have a rooted device, you can try Complete Linux installer. There used to be a project Ubuntu for Android, but since it's no longer supported, I can't recommend it.
Assuming a rooted device, you can also install a Debian or Ubuntu system manually. I described the procedure on Unix & Linux Stack Exchange.
¹ SysV IPC