To get started you will want to search on "divider circuit", "frequency divider", "frequency synthesizer", "counter circuit", "divide-by-N circuit" and maybe "flip-flop". Also, take a look at a data sheet for a "74HCT93" integrated circuit. Search on www.digikey.com for this. It is a very simple "4 bit binary counter" which will divide an input square wave by 2,4,8 & 16.
Be aware that most divider circuits operate on logic level signals. These are square wave signals by definition. So, they are of limited utility. Dividing other wave shapes and maintaining their shape is a much more difficult design task, something you will need a DSP (Digital Signal Processor ) or a PLL (phase-locked-loop" ) to perform.
Even in the realm of logic level dividers (e.g. the 74hct93 mentioned above) there are hard limitations of functionality. For example, there are many ICs available to divide by powers of 2 (2,4,8,16,32,64, etc.), and a few for non-binary integer numbers( e.g. 7490 on DigiKey) . Dividing by a non-binary number (3,5,6,7, 4922, etc.) is the domain of a CTC (counter timer circuit). CTC's (often simply called "timers") are included in most micro-processor ICs. There was a time when they existed as individual ICs, there may still be some out there.
Consider yourself "guided". Good Luck.