When to use a 1/4 Watt resistor vs 1/2 Watt resistor
The watt rating tells you how much current you can run through the resistor before it overheats and burns up. Power is given by:

$$P = I^2R$$

So the maximum current through a 1/4 watt resistor is:

$$I_{1/4W} = \sqrt{\frac {1} {4R}}$$

while the maximum current through a 1/2 watt resistor is:

$$I_{1/2W} = \sqrt{\frac {1} {2R}}$$

You can also use voltage instead of current:

$$P = \frac {V^2} {R}$$

$$V_{1/4W} = \sqrt{\frac R 4}$$

$$V_{1/2W} = \sqrt{\frac R 2}$$

For example, if you have a 100-ohm resistor:

$$I_{1/4W} = \sqrt{\frac 1 {4 \cdot 100\ \Omega}} = 50\ \mathrm{mA}$$

$$V_{1/4W} = \sqrt{\frac {100\ \Omega} {4}} = 5\ \mathrm V$$

Try putting 10 volts across a 100 ohm 1/4 watt resistor. You'll quickly see it start to smoke. The smell is very distinctive. Just don't hold the resistor in your hand while you do this!