Computer Mission Impossible
The motherboard is THE most important element, period. It is what is responsible for connecting every other part together so that they can do meaningful work.
The Power Supply
The power supply unit is even with the motherboard, as it provides the power necessary to run every component.
The Central Processing Unit
The central processing unit, as it is responsible for performing any and all calculations required by the computer. (Though modern computers will push off graphical work to the graphics card, which is specialized for such operations. Before the advent of such cards, the CPU used to do this work also.)
Next is RAM memory, which allows the CPU to access much more data than its onboard cache alone. This will drastically increase the functionality of the computer, as it can now run larger programs and work on larger data sets.
Hard drive, Flash drive and CD/DVD
Then a generalized 'tier 3 data storage', which includes hard drives, flash drives, and CD/DVD drives. This gives the computer persistent storage to allow it to save data when it is not powered.
The above parts are what I would consider 'necessary' for a modern computer. After that, importance order is dependent on the purposes of the computer, and none of these following parts are considered necessary. For instance, it's not uncommon for network servers to be 'headless' , but they will have one or more network adapters.