While developing the Roboduino, Scott had to calculate how many amps his project could deliver to attached motors. He wrote up his findings (and how he arrived at them) as a helpful article –

As a review, the total heat generated by the regulator is just the energy in minus the energy out. This is how linear regulators work–they turn excess energy straight into heat.

heat generated = (Vin-Vout)*I

(there’s also quiescent current in there which is the current used by the regulation circuity, but this is around 50mA, so we’ll ignore it for brevity).

So, if Vin = 9V, Vout = 5V, and I = 1amp, the heat generated would be about 4W.

As it turns out, our board was only able to dissipate about 2W. A bundle of 20 1/4W resistors (100ohms, 10ohms total) was used as a dummy 1Amp load at 5V. A thermal couple measured regulator temperature at the solder joint. Current was measured before it entered the Roboduino’s input.

Plus the article continues to explain how a circuits trace width determines how much current it can handle. – PCB as a Heat Sink

[Updated – Thanks, anachrocomputer!]