Why do we care about voltage and current not power only
For example: a red LED needs 2V to function, but why? why don't we say the LED function at - just assuming - 10 W, whatever the combination of voltage and current?


Because you can achieve 10 W in an infinite number of voltage and current combinations. e.g., from \$ P = VI \$ we can see that we could generate 10 W using


1 V x 10 A,
10 V x 1 A, 
100 V x 0.1 A,
1000 V x 0.01 A.




Figure 1. A 55 W headlamp bulb - but for what voltage supply?

In designing a device we have to make some decisions. Let's take the example of a 55 W auto headlamp bulb: somebody has decided that the standard battery voltage is now 12 V for cars and 24 V for trucks and buses. That means that the 12 V bulb will require 4.6 A and the 24 V lamp will require 2.3 A.

If you are designing a product from scratch then you have many options.

The LED example in your question is rather interesting in that the forward voltage, VF, is determined by the semiconductor used to generate the light. 



Figure 2. The IV curves for typical small red, orange and green LEDs. Source: LEDnique - parallel LEDs.

Note that the curves in Figure 2 are a fixed property of the material used to dope the semiconductor to get the desired colour. You can't change it.



From the comments:


  ... why normal diodes need 0.7V to work?


Because that's the voltage required to enable charge carriers to jump the depletion zone in the P-N diode junction.


  A p–n junction diode is made of a crystal of semiconductor, usually silicon, but germanium and gallium arsenide are also used. Impurities are added to it to create a region on one side that contains negative charge carriers (electrons), called an n-type semiconductor, and a region on the other side that contains positive charge carriers (holes), called a p-type semiconductor. When the n-type and p-type materials are attached together, a momentary flow of electrons occur from the n to the p side resulting in a third region between the two where no charge carriers are present. This region is called the depletion region because there are no charge carriers (neither electrons nor holes) in it. The diode's terminals are attached to the n-type and p-type regions. The boundary between these two regions, called a p–n junction, is where the action of the diode takes place. When a sufficiently higher electrical potential is applied to the P side (the anode) than to the N side (the cathode), it allows electrons to flow through the depletion region from the N-type side to the P-type side. The junction does not allow the flow of electrons in the opposite direction when the potential is applied in reverse, creating, in a sense, an electrical check valve. Source: Wikipedia Diode [emphasis mine].


0.7 V is sufficiently higher.