Question: I'd like to buy new resistors for breadboarding with my Arduino Uno. There are 0.25W, 0.4W, 0.6W, 1W, 2W, 3W, 5W, 10W resistors in the shop. What is the most suitable for this operation?

Answer: 
The answer is: It depends what you want to use them for.

You need a wattage that is higher than the power it will dissipate. What that power is depends on what you are doing with them. There are standard formulae that tell you the power in any given situation.

For instance, if you know the current and the resistance you can use: P=I²×R. If you know the voltage across the resistor it's P=V²÷R.

Take as an example the common resistor for an LED. Assume a 2V LED and a 220 ohm resistor. With a 5V total voltage the voltage across the resistor would be 5-2=3V. So the power would be (3²)/220 = 9/220 = 0.041W so you could quite happily use a 0.125w (⅛w) resistor for that job.

To be honest though most common Arduino jobs can be accommodated by ⅛W or ¼W resistors. ¼W is better for breadboarding since the leads are fatter than ⅛W, but ⅛W can take two leads at once direct into an Arduino's headers. Also if you fold the leads in half and squash them with pliers they fit better.
The smaller the better for the most part. On 1/4 or 1/8 W resistors the leads size is small enough to work well in bread boards. Rarely do you need to dissipate large amounts of power so 1/8 is a good size for logic, maybe 1/4-1/2 for power supply areas. See Majenko's answer for the heavy math, mostly applicable in power supply areas as he points out.