When you study electronics the first thing you learn is [u]Ohm's Law[/u] which describes the relationship between voltage, resistance and current. (BTW - "I" represents current, and I don't remember why.)
Resistance is "the resistance to current flow". More voltage = more current, less resistance = more current.
We usually don't know the resistance (or impedance) of the Arduino, or a motor, etc., and it can change depending on how many LEDs are on, or the load on the motor, etc. But, we often have a maximum current spec for the device. Ohm's Law is a law of nature, and it's always true but it's not always easy to apply directly.
And there is the old water-flow analogy - Voltage is similar to water pressure, electrical current flow is similar to water flow, and resistance is the resistance to water flow. i.e. A fat pipe has low resistance and lots of water flows. A skinny pipe has high resistance and less water flows.
But, there are a couple of big differences - If you cut a water pipe you get zero resistance and water flows-out everywhere. If you cut a wire you get infinite resistance and no electrical current flows. And, zero resistance in a "water circuit" doesn't cause any problems. Zero (total) resistance in an electrical circuit is a "short circuit" and stuff gets fried!
This is over-simplified, but voltage tends to be constant. If you live in the U.S. there is 120VAC at the wall outlet all of the time, whether anything is plugged-in or not. If you plug-in and turn-on a light, current flows. If you plug-in a hair drier or toaster, more current flows. If you plug-in two toasters, too much current will probably flow, the circuit breaker will blow, and the voltage drops to zero until you reset the breaker.
"Constant voltage" really means that the voltage is (largely) independent of the load... If you write a "1" from an Arduino output pin, you get approximately 5V and that doesn't change if you connect an LED with the appropriate current-limiting resistor.
If you try to get too much current from your 9V power supply, the voltage will drop and unpredictable things can happen... Maybe an internal fuse will blow, or something will burn-up inside. Or maybe there will be no permanent damage and everything will return to normal you remove the load.