trying to understand Bose-Einstein Condensate (BEC) I am a computer scientist interested in network theory. I have come across the Bose-Einstein Condensate (BEC) because of its connections to complex networks. What I know about condensation is the state changes in water; liquid-gas. Reading the wikipedia articles on the subject is difficult, not because of the maths, but the concept doesn't seem to be outlined simply. Other source share this approach going straight into the topic without a gentle introduction to set the scene for the reader. 
I would appreciate a few paragraphs describing the problem at hand with BEC (dealing with gas particles right? which kind, any kind? only one kind? mixed types of particles? studying what exactly, their state changes?), what effects can occur (the particles can form bonds between them? which kind of bonds? covalent? ionic?), what do we observe in the BEC systems (some particles form many bonds to particles containing few bonds? The spatial configurations are not symmetric? etc), and what degrees of freedom exist to experiment with (temperature? types of particles? number of particles?) in these systems.
Best,
 A: Each particle can be described entirely by its quantum mechanical state (1), which are a set of properties which distinguish one particle from another. More precisely, a quantum mechanical state is a particular combination of values for these properties. (i.e. if for two particles, these properties all match, they are in the same state). For fundamental particles (i.e. electrons), the state is the only way to distinguish the particles.
Associated with each quantum mechanical state is an energy which can be calculated if we know the state. The state in the system that has the lowest energy is known as the ground state. 
When a significant number of particles co-exist in the ground state we have a Bose-Einstein condensate.
Low temperatures are generally required for the ground state to exist since temperature imparts energy to particles, and so 'kicks' them out of the ground state into a state with a higher associated energy.
Briefly looking at the wikipedia page on this topic, it seems that the network theory analogy is concerned with the 'particles' of a system going from having a large range of properties to some number sharing the same properties, i.e. condensing into a single 'state'
An example given is a traffic jam. Before the cars hit the jam, they would have had a range of speeds, once the cars hit the jam, they have zero speed.

(1) Unfortunately the word state has two different meanings, the quantum mechanical state described above and one that is used in place of phase e.g. solid, liquid, gas etc.
A: First and foremost, the BEC systems studied in detail today do not involve the formation of any bonds between atoms. Bose-Einstein Condensation is a quantum statistical phenomenon, and would happen even with noninteracting particles (though as a technical matter, that's impossible to arrange, but you can make a condensate and then manipulate the interactions so they are effectively non-interacting, and the particles remain a condensate). 
The "high school physics" version of what happens at the BEC transition is this: particles with integer intrinsic spin angular momentum are "bosons," and many of them can occupy the same energy state. This is in contrast to particles with half-integer spin, such as electrons, termed "fermions," which are unable to be in exactly the same quantum state (this feature of electrons accounts for all of chemistry, so it's a Good Thing). When we talk about a confined gas of atoms, quantum mechanics tells us that we must describe it in terms of discrete energy states, spaced by a characteristic energy depending on the details of the confinement. Because of this, the two classes of particles have very different behaviors in large numbers.
The lowest-energy state for a gas of fermions is determined by the number of particles in the gas-- each additional particle fills up whatever energy state it ends up in, so the last particle added goes in at a much higher energy than the first particle added. For this reason, the electrons inside a piece of metal have energies comparable to the hot gas in the Sun, because there are so many of them that the last electron in ends up moving very rapidly indeed.
The lowest-energy state for a gas of bosons, on the other hand, is just the lowest-energy state available to them in whatever system is confining them. All of the bosons in the gas can happily pile into a single quantum state, leaving you with a very low energy.
It turns out that, as you cool a gas of bosons, you will eventually reach a point where the gas suddenly "condenses" into a state with nearly all of the particles occupying a single state, generally the lowest-energy available state. This happens with material particles because the wave-like character of the bosons becomes more and more pronounced as you lower the temperature. The wavelength associated with them, which at room temperature is many times smaller than the radius of the electron orbits eventually becomes comparable to the spacing between particles in the gas. When this happens, the waves associated with the different particles start to overlap, and at some point, the system "realizes" that the lowest-energy state would be for all the particles to occupy a single energy level, triggering the abrupt transition to a BEC.
This transition is a purely quantum effect, though, and has nothing to do with chemical bonding. In fact, strictly speaking, the dilute alkali metal vapors that are the workhorse system for most BEC experiments are actually a metastable state-- at the temperatures of these vapors, a denser gas would be a solid. They form a BEC, though, because the density of these gases is something like a million times less than the density of air. The atoms are too dilute to solidify, but dense enough to sense each others' presence and move into the same energy state.
The underlying physics is described in detail in most statistical mechanics texts, though it's often dealt with very briefly and in an abstract way. There are decent and readable descriptions of the underlying physics in The New Physics for the Twenty-first Century edited by Gordon Fraser, particularly the pieces by Bill Phillips and Chris Foot, and Subir Sachdev.
A: I think that you are asking specific particular questions to what I only know as a general quantum physics concept of a condensate. The other answers that I saw here, while looking myself for more details, all deal with particular cases of the general condensate concept.
As far as I know the BEC concept refers to specific equations of a function between a field's strength or a field's intensity and the corresponding energy involved. There are a few, -at least a couple A) and B) below- critical things to consider here for this to be a condensate:
A) The energy values can only be discrete, quantized, integer multiples o, +1, -1, +2, -2, etc. of a quantum "building block" value, AND:
B) You can add or take away "occupants" of the condensate without making a difference. My understanding of "occupants" are particles in the general sense of quantified energy associated with the vibrations of the field's intensity. Some particular examples have been mentioned in the other answers. I would add the example of superconductivity with electrons as particles and with energy in the electrical field.
Other useful related concepts are the fact that energy wave functions within a multi dimentional field can yield specific particle properties and interactive behaviors involving them, such as spin, angular momentum, charge - also charge other then electrical -, properties that for a condensate  have something to do with the spontaneous breaking of symmetry or at least the math's reconciliation in interactions with particles to / from a condensate. Also too I think that my statement B) above is the actual equivalent to having a statistically large number of particles in the lowest allowable quantum state. How this relates to Pauli's exclusion principle, or not, might be another subject.
From this general mathematical concept particular tangible applications can be derived such as your complex networks case  and  matter, energy, and force behaviour can also be explained such as the difference between fermions and bosons so well described here by Chad Orzel. As he also stated, when applied to elementary particles this has nothing to do with chemical bonding but it is rather a mathematical model of how matter, force, mass, energy, space, and time behave at the smallest tiniest level we can pretty much comprehend for now. A bunch of it has been proven experimentally but the jury is still out on some of its aspects as currently theorized.
