Resistors are used to
- limit current,
- devide voltage
- protect against over voltage
- pull-up, pull down
- current to voltage conversion
- etc ...
1) limit output current, the absolute max current per IO is 40mA, a typical LED works on ±2V 20mA.
the resistance value can by calculated by (5V - 2V)/(0.02A)=150Ω usually a 220Ω resistor is used, because: it consumes less power, there doesn't flow 20 milliamps, and there is no notable difference in emitted light.
2) if you have a analog voltage that variates between 0 and 10 Volts, you 'll need a voltage divider of 1/2. pick by example z2 10k and calculate z1 by 10k*(Vin,max/5V -1). take a value of resistance higher than the original calculated. and recalculate the new Vout.
3) place a resistor of 10k in series between the analog input of the arduino and the 'to measure voltage'
3) if you have to measure a analog current, you place a resistor to ground and the analog input, calculate the resistor by Z=5V/amps.
4) if you connect a button to the arduino, you 'll need to place a pull up or a pull down resistor. if you 're not using a resistor, the input is floating and can take any value (high or low). or you can enable the internal weak pull up. by pinMode(xx,INPUT); digitalWrite(xx,HIGH);. and disabeling by digitalWrite(xx,LOW); by default the pull-up is disabled.