How do devices/appliances draw more current when needed?

I decide to do something that requires more processing power. In this case my computer will draw more current to increase power.

Other way round: the computer will do more things, and as a result it will consume more power.

Does my computer open up more parallel circuits so the total resistance decreases?

This is roughly true. Except that computers don't really operate on continuous current flow, they operate in bursts driven by their internal clock; each action involves either drawing some current to turn on a transistor, or sinking some current to turn it off again. Times a billion transistors, a billion times a second. More computation involves more transistors.

At a high level, yes, you're right that the computer opens up more transistors or at least switches more transistors when it consumes more current. For instance, if you have a hardware multiplier and generally you don't use it, the transistors in the multiplier won't switch on and therefore won't draw much current. If the code now requests a multiplication, the transistors in it start switching and that will lower the resistance between VDD and ground. This will draw more current. The current draw will lower the VDD voltage. Now the switching voltage regulator will detect this voltage droop and kick on at a higher duty cycle to allow high current capability and approximately a constant voltage.

At a broad high-level, circuits request more current by lowering their resistance because most circuits operate with a constant voltage source.

Modern computers use logic gates that are designed to use very little power when they are in a steady state, but which take a burst of power to switch them from one state to another.

If the computer is idle, the processor will be in a sleep state for much of the time. Most of the circuits will be doing nothing, and so consume little power. The same goes for other components, such as the graphics card's GPU.

If you then give it something to do, then suddenly it's performing more work. The gates are switching on and off more often, and so they take more power.

In addition, many computers, especially laptops, are designed to power down whole sections of the computer if they are not being used. For instance the webcam in a laptop will be powered off until you open an application that uses it.