I confess that I do not share your difficulty with the concept, in some measure, perhaps, because I view it mathematically.
Others have discussed entropy as a philosophical concept, and Dr. Stochastic refered to it as a thermodynamic state variable (which it is). However, I prefer to view it from its original derivation in Statistical Mechanics. I always found Thermodynamics hard but Stat Mech easy.
You are indeed correct, that the first law of thermodynamics states that energy is conserved. Therefore, the total heat and work are conserved quantities.
The second law, again you are correct, addresses the distribution of energy.
From its original, mathematical derivation, entropy is indeed a measure of how energy is distributed. It is the natural log of the function omega. Omega is called the density of states function, and gives the total number of quantum states available to the system.
It is also called the partition function, because it defines, essentially, how the energy is partitioned among the various quantum states. In that sense it is usually considered as a distribution function. Indeed, in gas and plasma physics, the derivation is reversed. Assuming a maximum entropy state (i.e. equilibrium), and given a temperature and pressure, you can than calculate the partition of energy among independent, free particles (therefore they do not have discreet quantum states per se). The result is the Maxwellian distribution, which is the three dimensional velocity distribution, or the Boltzmann distribution, which is the 1 dimensional energy distribution. These distributions are the equivalent to the partition function in that they give the distribution of energy among the population of particles. It is a fundamental concept in physics. Systems that are defined as "thermal" are those for which the particle distribution (in energy or velocity) follows the Maxwellian. It is the basis for the thermonuclear bomb and for the definition of temperature.
Thanks so much for this information, 2ndreconmarine! Fine post!