I have a Huzzah ESP826 Wifi breakout chip which requires 3.3V power voltage and peaks at 500mA of current, and an LED strip which requires 12V and 1A. How can I power both with one dc jack (I have a 12V 5A power supply) on one breadboard?
You need a voltage regulator for the esp. Preferably a dc-dc step down, which will neither waste power nor heat up. Other, passive solutions are possible but not desirable. The regulator can be found on ebay amazon etc.
One more thing about the LEDs: depending on the model you have, they might or might not work with 3.3V input (I'm assuming you want to drive the LEDs from the esp).
In case they take only 12V input, you will need some interface solution, like this one, for example.