ADB is the acronym for Android Debug Bridge, which is part of the Android SDK (Software Development Kit). It uses a client-server-model (i.e. adbd, the ADB daemon, is running on the device and can be connected to), and in most cases is used via an USB connection. It is also possible to use it via WiFi (wireless adb).
There's nothing you need to install on your Android device, as the ADB daemon (adbd) is already integrated into the Android OS. It is usually accessed via a command line interface from the PC, where either the full Android SDK is installed (several 30 MB download archive currently), or a massively stripped-down version for "non-developers", sometimes referred to as "Mini ADB" or "ADB essentials" (for Linux, this is only the adb executable; for Windows it's adb.exe plus two or three .dll files).