Console firmware is effectively the operating system of the console. It’s called firmware because it’s “baked in” to the hardware and is more difficult to change (“firm”) unlike say a PC or application that can be modified at will (“soft”).
CFW stands for Custom Firmware, it just means any firmware for a device like the Nintendo DS that is supplied by someone other than Nintendo. CFWs can do all sorts of things, it depends on the exact firmware. But it might be common to find firmware that allow easily running homebrew software (DS software that is not officially sanctioned by Nintendo), or provide features that you might find on a typical operating system like file browsing, text editing etc.
A typical DS game cartridge uses Read Only Memory, meaning the game cannot be modified after manufacture. A flash cart is a third-party device that emulates the functionality of a real DS cart, but allows the user to easily modify the data content (eg. with a micro SD card). Often they are used for storing one or more (pirated) game ROMs, or homebrew software. Sometimes flash carts may also be used as the delivery mechanism by which to install or run a custom firmware.
So, DS CFW is a custom operating system for the DS console, DS flash cart is a hardware device to load custom software on a DS, which may require a CFW, or be used to load the CFW itself.
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