Why do we need constants? This question is driving me crazy because I cannot find a straightforward answer. I want to know what a physical constant exactly is. I know that it’s a value that doesn’t change, but what is it? Why do we need them in our equations?
For example what is the gravitational constant? Why do we need it in the universal gravitation formula? What is Planck’s constant? I don’t want to know the value, I want to what its use is. What does it allow us to do or find? What exactly does Planck’s constant tell us?
 A: Physical constants arise from the way we define units. Let's take the gravitational constant $G$ as an example. According to Newton's law of universal gravitation:
$$F_{g} = G \frac{m_1 \times m_2}{r^2}$$
If you were to take two spheres, both with mass 1 kilogram, 1 meter apart, it turns out the gravitational attraction between them is not 1 newton: it would be $6.674 \times 10^{-11}\ N$. Thus, the gravitation equation needs a conversion factor with value $6.674 \times 10^{-11}$, which we call $G$. We need it just because of the way our unit of force, the newton, is defined: 1 newton is defined as the force needed to give a mass of 1 kilogram an acceleration of 1 meter per second squared. 
The kilogram, meter and second are rather arbitrary quantities, and as such, the newton is an arbitary unit. We could have equally well chosen to use yards instead of meters, and pounds instead of kilograms, and define the newton as the force needed to accelerate a mass of 1 pound by 1 yard per second squared. This would give us other values for the physical constants. Nature, of course, couldn't care less how we humans define our units. Nature has it's own units.
We need the physical constants to convert the effects of nature into the units of our choice. The gravitational constant $G$ converts the gravitational force between masses (in kg) seperated by some distance (in meters) into Newtons. Planck's constant $h$ converts the energy of a photon with some wavelength (in meters) into Joules.
A: Actually, we do not really need constants in the sense that we choose to use them, but it’s just the way the universe works – or more precisely: very well seems to work.
For example, various experiments confirmed that the quotient of the energy of a photon and its frequency (when measured in the same units) is always the same within the accuracy of what can be measured. We did not chose the universe to be this way; it just is. In the above case, we agreed to call the quotient $h$ and we usually also agree on some units to communicate the value of this quotient. This in turn allows us to make accurate predictions about reality – in this case, what energy a certain photon has.
Now, the whole unit concept is again based on constants and, once more, they are brought upon us by the universe, which happens to be very well described by numbers. For a blatant example, suppose we take two blocks of wood and cut two pieces of string such that they just begin and end where one of the blocks begins and ends. If we then tie those strings together, we find that the composite string begins and ends where the two blocks of wood begin and end, when put together. This description is awfully complicated, because I tried to avoid the word length, which denotes the fundamental property of objects derived from such observations and which eventually lead to the invention of mathematics to better describe such properties. 
However, for being able to talk mathematically about specific lengths without resorting to actual pieces of string, we need to agree on a unit length, which is nothing but an arbitrarily chosen constant to facilitate communication. While we chose what to use as units, we did not really chose to use units in general: The universe just happens to be such that they are damn useful and without them, we would not have the time to think about such questions.
A: Many of the units related to physical phenomena were defined before the phenomena in question were particularly well understood.  For example, electric current was measured based upon its magnetic effects before it was understood that the amount of current called "1 ampere" represented the flow of some number of electrons per second, and "2 amperes" represented the flow of twice as many electrons per second.  Since the units were defined before there was a means of quantifying the number of electrons per second they should represent, the number of electrons per second representing one ampere ends up seeming rather arbitrary.
If one had the luxury of defining units without regard for earlier practices, it may be possible to assign them so that most ratios of interest would have a scaling factor of one (e.g. if time is measured in units called the "jiffy", one could define unit of distance called the "bip" equal to the distance light travels through a vacuum in one jiffy, then the speed of light through vacuum would be one bip/jiffy).  From a practical standpoint, common units would need to be scaled by powers of ten (e.g. if the speed of light is one bip/jiffy, expressing a typical highway speed limit as 0.0000001 bip/jiffy would be awkward, but 100 bips/nanojiffy might be less so).
Even if one had the luxury of defining units, however, it would probably be impossible to avoid having some non-unity constants slip in.  The constant 2pi, of course, would be one that would arise since something that travels at a 1 bip/nanojiffy around a circle with a 1 bip radius will have a rotational period of 1/2pi nanojiffies since the distance per revolution will be 2pi times the radius.  Further, it's convenient to have units which are defined according to some properties of the big rock called Earth.  Having the basic unit of time be an integer fraction of the "aparent" rotational period of the Earth relative to the Sun, for example, is generally convenient [though there could arguably be advantages to having a "scientific" time unit which not a convenient multiple of to civil time units derived from the Earth's rotation, since it would mean that "scientific" time could run uniformly without any need for "leap seconds" even if the Earth's rotation speed varies slightly.]  Having the earth's rotational period divided into 24x60x60 seconds is more convenient than would be, e.g. having it typically take about (made-up number) 4,291,269 jiffies.
