the difference between the 1/4W and 1/8W type is the maximum power which the resistor can stand. the electrical power is amperage * voltage. so in the case of a 1k resistor connected between an arduino output and ground:
5V is the output voltage of arduino, so the amperage is 5V/1kohm = 1mA. so the power is 5V1mA = 5mW (VI = Power -> V*V/R = Power). the 1/8W would be strong enough to stand this.