Why do computers have clock signals and clock rate, while ordinary machines do not have them?
And why is clock so fundamental in CPU and mainboard?
"Time is what keeps things from happening all at once." - John Wheeler (1911-2008)
You don't want to run things unorganized. Everything in a microcontroller happens consecutively, in discrete steps. Getting a byte from RAM to a register: put address on address bus. Tick. Assert RD line. Tick. Latch data from databus into register. Tick.
If you would do this all at the same time, you would latch the data before the RAM had time to put it on the bus. This kind of limitations occur all the time, and they are the limiting factor for a controller's performance.
Note that some mechanical systems use a mechanism with a similar function:
A watch's escapement also ensures that things happen timely, and even in an analog clock, in discrete steps. (Personally I count the escapement, along with the zipper and the wood screw, to some of the Truly Great Inventions.)
Most machines require that some types of operations happen in some particular sequence. A gear train, for example, requires that each gear move in such a fashion as to make room for an approaching tooth, before that approaching tooth arrives. In many cases, such as with the gears, the part of a machine which requires certain events to happen in a particular sequence will be very tightly coupled to the part which enforces that sequence (e.g. the proper interleaving of one gear's tooth with another gear's groove will be enforced by the adjacent teeth, which are of course rigidly attached).
What makes computers different from other machines are the facts that (1) there are a lot more parts that have to be synchronized than is typical of most machines, (2) the maximum speed at which parts of a computer can be driven is faster than the maximum speed at which they will function reliably, and (3) it is often desirable to run computers at the maximum speed where they will function reliably.
The easiest way to satisfy all of the above requirements is to design parts so that they will function reliably if run at some particular speed, and then run them at that speed. It's helpful to think of a computer as an assembly line. The number of products produced per hour will be set by the rate at which they conveyor moves products from one station to the next. Using a conveyor whose speed is the same throughout a factory will ensure that no workstation will have product arrive before there is space available. Increasing the speed of the conveyor will increase the rate at which products are produced, provided that each worker is able to complete his part of the production in time. If the conveyor is sped up too much, however, workers may sometimes fail to complete their tasks properly, thus creating product defects. Note that if one particular step in the process takes longer than two other adjacent steps elsewhere in the production, it may be helpful to either split the longer step into two steps (which would require more workers, but allow increased production), or join the shorter adjacent steps into one (which would not increase production, but would reduce the number of workers required to achieve it).