How can quantum entanglement be explained in layman terms? I was just wondering if one entangled particle was able to influence its partner, because they both remain connected through the electromagnetic field. 
e.g via the same field line, so that either information can be sent along the connecting field line or that the field line only allows certain values for particles (local field excitations) that are entangled.
Could disrupting the electromagnetic field break entanglement? e.g. sending one of the entangled electrons down a black hole.
 A: Honestly, I think you cannot explain entanglement "in layman's terms" without introducing a totally erroneous conception of it.   Entanglement is NOT a physical link like you suggest.  Entanglement is a direct consequence of the cornerstone of quantum theory, which is the superposition principle, and that principle, by itself is so "weird" in layman's terms that you risk to convey a wrong idea, that you will get insulted as being a madman, or that you will be thought of as talking nonsense.
Here is the superposition principle: if $A$ is a possible reality, and $B$ is a possible reality, then $a A + b B$ is also a possible reality, with $a$ and $b$ complex numbers.   And here is the Born rule: if $c C + d D$ is the reality you're measuring, then the probability that you will actually "see" $C$ is proportional to $|c|^2$ and the probability that you will actually "see" $D$ is proportional to $|d|^2$.  From there on, if you saw $C$, you can assume that the reality you'll be dealing with from that point on, is $C$, and if you saw $D$, you can assume that the reality you'll be dealing with from that point, is $D$.
That's the shortest intro to quantum theory I can think of.
Entanglement is a consequence of that.  One reality is that particle 1 is in state $U$ and that particle 2 is in state $V$.  We'll write that reality as $\left|U,V\right\rangle$.  Another reality is that particle 1 is in state $W$ and particle 2 is in state $X$.  We'll write that reality as $\left|W,X\right\rangle$.  Well, the superposition principle tells us that realities of the form $(a\left|U,V\right\rangle + b\left|W,X\right\rangle)$ are also possible, and if we're in such a reality, we say that particles 1 and 2 are "entangled".  They don't have a specific individual state, because the actual reality is composed of different combinations.  So if we do a measurement, we'll have a chance that we'll find a reality $\left|U,V\right\rangle$ with probability $|a|^2$ ; but we might also find $\left|W,X\right\rangle$ with probability $|b|^2$.  However, we can't find, say, $\left|U,X\right\rangle$, because that wasn't part of the reality when we started measuring.
However, I don't think that the above qualifies as "layman terms".  I don't know how to make it simpler, and I don't think you can without killing the (very very weird) concept that's behind it: the superposition principle.
A: Let us make this clear. The word entanglement in webster dictionary

1a : the action of entangling : the state of being entangled
1b : something that entangles, confuses, or ensnares
2: the condition of being deeply involved ,    their entanglement in politics

has been taken over by quantum physics nomenclature, or the sense of 2
There is no "action"  as you describe.
You are thinking in terms of classical physics, but the word has been used to describe  quantum mechanical observations and solutions of quantum mechanical equations .
You could use the word for classical physics : If you solve the gravitational equation for a planet with a satellite revolving around it  the total angular momentum of the system is zero, if  then  you measure the angular momentum of the satellite, you do not have to measure the angular momentum of the planet, because of conservation of angular momentum, you know immediately that  it must be the negative.
The word  is not used in classical physics because it is so evident a situation and it is not necessary to give a particular name to the condition.
Quantum mechanics has very well defined equations which describe the dynamics of quantum systems , they predict probabilities for measuring particles at specific locations with specific energy and momentum , after an interaction, not the exact values describing classical physics interactions. Nevertheless, conservation laws are strict and also apply to quantum mechanical solutions.
In a quantum mechanical solution for a specific problem there are a number of conservation laws in addition to energy, momentum and angular momentum, there are the conserved quantum numbers.
Two particles decaying from a quantum mechanical state with total spin zero by conservation of angular momentum will have to have equal and opposite spins, as with the decay of a pi0 into two gamma, the pi0 having spin zero and the gammas each carrying spin one, +1, or -1 in projection on their direction of motion. No matter how far away one then measures one of the gammas spin projection, the spin projection of the other is immediately known. No probabilities because it is just a conservation law.
This needed no special word to describe it until the advent of the field of quantum computing , where many particle systems are set up in a quantum state , they needed a simple word to describe the effect on the bits they were using as bits in the computation set  up. They called "entangled" the particles that by conservation laws have to carry quantum numbers uniquely correlated to each other by the quantum mechanical solutions. So if one measures the values for one, the values for the other is immediately known.
No action has to take place for this knowledge to be transmitted because it is in the fixed quantum mechanical  mathematical solution.
Breaking entanglement means that a measurement was taken, new interactions, and the quantum mechanical solutions have new boundary conditions. In the planet example above it would mean an asteroid hitting the planet and bringing new angular momentum in, changing the boundary conditions .
A: I am very fond of a particular game for explaining entanglement. It is a cooperative game: three people are on a team and they're trying to beat the game itself. I have talked earlier on this site calling the game "Betrayal," it is a game that tries to force one of the people to betray the others.
You put these people in three isolated rooms each with a screen and two buttons labeled 0 and 1. You flash some instructions, then they have 5 minutes to press exactly one of the buttons once, then you collect these three numbers together and add them up, call it "the sum." They win if the sum is even or odd, depending on what we ask them to do.
After they're isolated we choose one of four experiments at random. One quarter of the time, we run a "control experiment", where we display on all of the screens "make the sum of your numbers even," and they win if the sum is even. That's the easy case where there is no traitor.
The other three quarters of the time, we choose one of the three at random to be the traitor. We display on their screen the wrong instruction, "make the sum of your numbers even," and we display on the other two screens the right instruction, "make the sum of your numbers odd." They win if it's odd.
Classical players of this game can only satisfy 3 of the 4 cases at any given time, no matter how complicated their strategy was. So their maximum success probability is 75%. Quantum players of this game are, surprisingly, able to beat it 100% of the time in theory -- except that quantum states are very fragile and so it is very hard to get them to last long enough to make this game happen. But we could say "you're going to play this game 100 times and you must win at least 90 of them" and a classical team with a 75% success rate would only win one in about 7000 of these while a quantum team with a 95% success rate would only lose one in 90. So you can give a quantum team a little leeway for their systems to be imperfect while still telling the difference between these two groups fairly accurately.
The single most important things about entanglement are carefully exposed by this game:


*

*Entanglement is about strange correlations observed when we bring information back together into one place. No individual room sees the entanglement directly; they adjust their quantum apparatus and half the time it tells them to press 0 and half the time it tells them to press 1 and they don't know anything more. But we see this entanglement feature when we bring back their digits and sum them up.

*Entanglement resembles what classical agents can already do, but it can achieve correlations which they couldn't possibly have made classically.
A: While entanglement is created through interactions of some kind (like EM interactions), it's a type of quantum correlation, which is an informational concept and doesn't need to relate to any physical substrate.
Think of the "correlations" between the two sides of a coin (if one is head the other is always tail).
Sure you need the coin to have it, but the concept of correlation itself doesn't, and can be proficiently studied without worrying about any physical implementation.
It's the same with entanglement.
The concept of quantum entanglement can be (and it very often is) studied without any reference to a physical implementation of it.
Entanglement is a "stronger" version of such classical correlations.
Two parties (particles, or whatever else) can be entangled regardless of their relative distance in space-time.
In principle, they can be one outside the other's light cone (though in this case it's impossible to exploit such entanglement, so it's useless to talk about it).
In principle, you can't break the entanglement between two distant parties, as local unitary operations cannot break entanglement.
In practice, it is extremely hard not to break it, and without extreme care, the "entanglement link" between two parties gets easily diluted making it impossible to exploit. Basically, any interaction with the environment will do.
"Sending one particle into another universe", whatever that means, I would say falls into the category of: even if there is still entanglement, its impossible to use, hence it's useless to talk about it.
A: A layman's explanation is going to be difficult.  One of the reasons you're asking this question is because entanglement is famously unintuitive.  I recommend Minute physics' video on Bell's Inequalities.  It's a really good video and it captures the unintuitive side of QM quite well.
What we will find, from these videos, is that the information cannot be transmitted via electromagnetism.  We've done studies which prove that any "signalling" which might be done between particles must be done faster than light (or some other mechanism beside naive "signalling" must be involved).  Your electromagnetic field simply cannot propagate the information fast enough to generate the effects we see in real experiments.
A: Entanglement is a logical relation between measurable parts of a single quantum state.
By "logical" I mean that it is not physical: it is a property of our mathematical representation of a physical system (this representation being the quantum state), but it is not associated to any conceivable aspect of the system itself.
By "measurable parts" I mean that the physical system described by the quantum state has distinct attributes (in the case of two entangled particules, it may be their positions, momentums, spins) that can be measured independently.
The "logical relation" that is entanglement makes the measurement outcomes dependent on each other: consistent sets of outcomes are expected, not justs any possible outcome for each measurement. The consistency can be understood as a conservation law. Thus the dependency is not a physical interaction, its origin lies in the symmetries of the world, which have to be respected.
The weird thing with entanglement is that although the "measurable parts" can be far away from each other and causally disconnected (no measurement on one side being able to influence the measurement on the other side), we still have random outcomes on each measurement (according to Born rule).
How can we accomodate the random nature of each single measurement outcome with the fact that both outcomes are still correlated? This is an open question. 
Some people think that a physical relation supplementing the logical one is missing (hidden variables), but this has been proven by John Bell to be very difficult to sustain. Other people think that Nature is non-local, meaning that in some way the spatial separation is not a separation at all, but this goes against the local nature of all known physical laws (and notably against the structure of the spacetime of general relativity). Yet other people have other ideas, that I will not discuss here.
A: So entanglement doesn't use a force or force field like the Electromagnetic field to maintain "connection" or entanglement.
This is my attempt at describing entanglement in layman terms:
First we have to describe some basics:
The state of a system in quantum mechanics is described by listing various "independent" observable outcomes it can have, and prescribing a probability to each of these outcomes (really a complex number whose squared magnitude gives the probability).  When there are multiple outcomes with non zero probability we say the system is in a super position of these outcomes.  For example a particle could have two outcomes:
spin up
spin down
if we assign $$1/\sqrt2$$ to both these outcomes, there will be a 50-50 chance of measuring the particle as spin up or spin down and thus we say it is in a superposition of being spin up and spin down.
This listing and prescribing of complex numbers is called the wave function of the system.  It describes the state of the system.  As the system evolves or is measured, the numbers(or probabilities) prescribed to the outcomes changes.
When a measurement is made, an observer finds a given outcome with the prescribed probability and the numbers change as a result of the measurement.  The number associated with the outcome becomes 1 and the rest become 0.  This is called wave function collapse.  It is a completely addition assumption added to the laws of physics.  It is in general considered a mystery why we must add this assumption.
Entanglement is all about measurement on a part of the whole system
We say that one half of the system is entangled with the other half if the wave function collapse changes the probabilities for the outcomes of the other part of the system.  Take for example a 2 particle system where each particle can be spin up or down, lets say 0 or 1 for simplicity.  There are then 4 independent outcomes:

*

*both up : 00

*both down: 11

*first particle up seocnd one down: 01

*and vise versa: 10

if all out comes are equally likely, we would perscribe the numbers$1/\sqrt4$ to each outcome:
$$00: 1/\sqrt4 $$
$$01: 1/\sqrt4$$
$$10: 1/\sqrt4$$
$$11: 1/\sqrt4$$
In this situation, if we ignored one particle the other particle has a 50-50 chance of being spin up or down.
When we make a partial measurement, we set all the independent outcomes incompatible with the measured outcome to 0 and multiply the rest by the a number (this is so they still describe a probability).
For example, in the all equal possibility, if we measure the first particle being down or "0", the system collapses to:
$$00: 1/\sqrt2$$
$$01: 1/\sqrt2$$
$$10: 0$$
$$11: 0$$
the probability of measuring the second particle with spin up or down is unchanged.  The all equal probabilities state is unentangled.
Now if we start with this state:
$$00: 1/\sqrt2$$
$$01: 0$$
$$10: 0$$
$$11: 1/\sqrt2$$
Then when we measure the first particle having spin-down ("0"), the wave function collapses to:
$$00: 1$$
$$01: 0$$
$$10: 0$$
$$11: 0$$
The probability of measuring the second particle as spin down has changed from 1/2 to 1.  By measuring the other particle, this mysterious collapse process determines the state of the other particle.  This change in probabilities due to measurement is called entanglement
