can one resister be used for multiple circuits?

Are these the same thing?

Is it possible to use one resistor for multiple circuits or must a resistor be used on each circuit? Also does it matter if the resister is on the positive or negative side ?

either one works it will just change how you go about calculating the resistance you need for the led. The resistor can be on the positive or negative side of the led as well both will work.

if you have a set current for the leds and set voltage being applied what you will see is the resistor on the left will be some value R and the resistors on the right they will each be R*3.

To use the same resistor for several LEDs is a BAD idea. The resistor is a current limiter and it will work but if one LED fail the other two will get a higher current and burn out over time. And if you want to learn more about resistors, https://en.wikipedia.org/wiki/Resistor