For example, in project #3,the love-o-meter, i had to divide the sensor value by 1024.0, and the explanation for it was never given in the manual.
The analog-to-digital converter in an AVR processor works by comparing the input voltage to a reference voltage divided into 1024 steps. It's called a "successive approximation analog to digital converter".https://www.google.com/search?q=successive+approximation+analog+to+digital+converterhttp://en.wikipedia.org/wiki/Successive_approximation_ADC
Furthermore, in project #4(which i'm currently at), the book states that i have to convert sensor readings from 0 - 1023, to a value between 0 to 255.Is the number 1024/1023 simply an arbitrary number/limit set by the sensor's manufacturers?
No. The value 1024 is the number of steps in the analog-to-digital converter and is independent of the sensor. 1023 is simply the maximum possible value from the analog-to-digital converter.
Also, is they any significance or reason that analogueWrite only accepts positive values up to 255?
Yes. The timers in an AVR processor can generate a pulse width modulation signal. For the Arduino, the timers are configured to run in an eight bit mode. The largest eight bit value is 255.