Incandescent bulbs will work fine with AC and DC power. All your household ligthbulbs were powered by 125V AC.
LEDs only light up when the voltage is higher at the anode then at the cathode. Only then the current will flow and the LED will light up. If you use DC power, and hook up the positive lead of the supply to the LED's anode and negative to the Cathode side (also using a resistor in series with the LED of course) then the current will flow and LED will illuminate. If you hook it up in reverse, the LED will not conduct any current and will stay dark.
AC power from wall-warts flips polarity 60 times a second (60Hz) If you use that power source to light up the LED, the LED will light up every time the AC polarity is correct for that LED circuit (which is 30 times a second). Because of persistence of human vision the LED still look like it is illuminated, but in reality it flickers 30 times a second. And since it is not lit 50% of the time, it will not glow as brightly as if it was powered from a similar voltage DCC supply. To demonstrate that the LED is flickering, take the lit LED in your hand and swing it left and right very fast. That will show you that the LED is not on all the time (you will be able to see the strobing effect).
I prefer to use DC power for the LEDs. One of the reasons is that white and blue LEDs have very low maximum reverse voltage rating. When the AC polarity is in the reverse cycle (the LED is not lit) the voltage might be exceeding the reverse voltage rating and the LED could be damaged.
