The video game systems or first generation consoles (before they were called consoles) from the late 1970s and still today are developed much like any embedded computer system.
An embedded computer is a computer (i.e. a microcontroller or microprocessor) in an electronic system that does not resemble a personal computer, or larger multi-user computer system (e.g. time-sharing system, mini-computers, and mainframes); one major difference being they typically lack of the now standard input/output via a video terminal and keyboard.
The development is done on another computer, these days typically a developer PC or workstation using a cross-platform development environment which produces executables for the target system - the video game system in this context, which is often be a different microprocessor architecture versus the development system's own (or host) CPU.
Two examples of current video game systems are the for Sony's Playstation 3 which is based upon the Sony, Toshiba and IBM's Cell processor, and for mobile devices (smart phones, tablets) which are often ARM based microprocessors.
Michael Madsen indicated the majority of the programming was done in assembly rather than a high level compiled language, such as C or Pascal. This was necessary to fit the game's entire content within the confines of the highly constrained system (very little RAM and limited cartridge ROM capacity), as well as to access co-processors such as the Atari 2600's TIA.