Difference between Arduino 5V power and output of 7805 5V power

You are going to have to post a diagram of how you wired everything up.

Also, when calculating the resistor for an LED, Vr=Vs-Vf where Vf=forward voltage of an LED, Vr = voltage across resistor, Vs=supply voltage.
A standard Red LED has If=20mA (forward current), Vf = 2.1v(forward voltage).

So how much voltage is across the resistor? (Hint: think of it like a staircase, the supply lets you walk up 5 steps, then the LED forces you down 2.1 steps, so how high are you?)
The resistor needs to have 2.9 volts across it to get you back down to the bottom of the staircase.

Its a series circuit, so all current that flows through the LED must flow through the resistor, meaning that there must be 20mA flowing through the resistor if you want 20mA to flow through the LED.

So using V=IR, you get 2.9 = 20mA * R => R = 2.9/0.02 = 145R. You will notice that there are some standard values for resistors, so the minimum standard value you would want is 150R.
You will find that an LED can run quite happily at a lower current, so you can use a larger resistor if you want, the LED just won't be as bright, but its working life will be extended.

For example, at 270R, you would have I = V/R = 2.9/270 = 10.7mA flowing through the LED.