Intuitive definition of electromagnetic field I am curious about the following:
Checking about what is an EM field in wiki it says:  

An electromagnetic field (also EMF or EM field) is a physical field
  produced by electrically charged objects   

Checking what is electric charge it says:  

Electric charge is the physical property of matter that causes it to
  experience a force when placed in an electromagnetic field

Is it my ignorance on the topic or is this a circular non-sensical definition? 
It seems to me after reading this, you don't know any more than after reading it.
Can we not define these in a "intuitive way" that is non-circular?
 A: Recall that science is descriptive of nature. These words are defined so that they describe the effects of experiments (generally the descriptions that stick are the simplest ones that cover the range of behaviors actually seen). 
As a result, though they are circular in structure, they absolutely do not rely on circular logic for their validity: they rely on empirical experience.
A: It's not actually circular, speaking purely logically.
One says "An X is a field produced by things having Y."
The other says "Y is a property of matter that causes it to respond to an X."
Putting these two things together reveals simply that some matter has Y, which produces an X, which affects other bits of matter which have Y's.
In other words we are describing some matter-matter interactions by putting a field between them, produced by the one bit of matter, to which the other bit of matter responds.  It's not nonsensical at all, and in fact it contains part of one of Newton's laws deeply within it: to cause such forces you have to be responsive to such forces. 
The only thing needed to make such a definition helpful is to then specify a known source of electromagnetic charge, like a proton or electron or rubbing wool on glass. Once you can reliably generate some matter which has Y, you can investigate Y in other matter. Until this is done, yes, the definition is a little over-broad because it does not pick out distinctively electromagnetic phenomena from all of the other phenomena that exist in this universe.
A: It is not acceptable to change a question to invalidate two existing good answers to your question, you should have asked a new question.
There are two types of particles in the universe, the fermions and the bosons. The fermions take up space in the sense that you can't have too many in the same region of (phase) space. And the bosons basically mediate interactions with the fermions. You can tell whether something is a fermion based on its spin. The Higgs boson has spin 0, the Z, the W+, W-, photon and all the gluons are spin 1 and the graviton (if it exists has spin 2). All integers. The fermions have spin that is not an integer. They include the leptons and the quarks all of which have so in 1/2.
Now we said bosons mediate interactions. So we have to say which things interact with each other. This is like saying who is married to each other. It is a grouping. It could be that everything interacts with the graviton (if it exists) or with the Higgs. But not everything interacts with the photon. All the quarks do but only half the leptons do. But the leptons interact stronger than the quarks do. So we assign a quantity to express this interaction. And this type of interaction just needs a single number (positive or negative) to characterize it. So we assign a charge to everything and assign a charge of zero to the leptons that don't interact with the photon and we assign a charge to the down quark but we assign three times that charge to the electron. And the -2 times as much to the up quark and -3 times as much to the positron. And so on. Because that is how much much stronger they react to photons.
But that was just one boson, the photon. We also need charges to describe interactions with other bosons. For instance the gluons need color charge which isn't a positive or negative number but a 3d space of colors and anticolors, it is a more complicated interaction. Even more so since gluons themselves have color charge.
Since we have different charges for different interactions we have to give them different names, so we call the charge for how strong you interact with the photon the electric charge. So we have electric charge and color charge. The quarks have both the electron, positron, muon, anti-muon, taon, anti-taon have electric charge but no color charge. The photon has neither charge. The gluons have a color charge (color and anticolor) but not electric charge. The W+ and W- have electric charge but no color charge.
This isn't circular because we haven't mentioned electric fields. We have lots of things that interact in lots of ways and so we have to say which interact in which ways. And rather than list them all we can say things like that the electron and a muon and tauon and W- all interact with the photon in the same way because we can assign them the same electric charge and then say how everything with that a charge interacts with the photon. And even better we can just say they interact three times as strong as the down quark (which has the same charge as the strange and bottom quarks and the anti-up, the anti-charm, and the anti-top quarks). This saves a lot of effort comparing to just giving the name of each particle and then saying how it interacts with the photon. So we assign a thing called charge to thibgs based on how they interact with a boson.
Now you might not be two surprised that photons and electric fields have similar jobs. Sometimes a bunch of photons can together have nice enough properties that you want to describe them with a simpler mathematics. The proper mathematics for photons is to have a bunch of operators at each point in spacetime. But sometimes describing a six dimension combination of planes in a 4d spacetime is pretty accurate for describing what it is and what it does. And sometimes we can pick a frame and describe that 6d combination of planes in 4d with two vectors. Though someone in a different frame would get two different vectors but they would both agree on the 6d object it formed in a 4d spacetime they really are just cutting up the planes in 4d into two groups in different ways, much like when you pick an x y and z axis you can cut up a line in 3d into a combination of numbers. Someone that picks different directions for their axis will get numbers but will agree on the 3d line in question.
OK, so after they broke the 6d combination of planes in 4d into two 3d vectors one of them is the electric field vector. It it one half of an approximation to how every photon in the universe is collectively net interacting with charges at that point.
And it turns out that when that interaction is behaving simply enough for this approximation to work the interaction ends up being as simple as making the thing with the charge feel a force proportional to the change.
Which is actually pretty simple since in general the interaction with the photon could be as complicated as orchestrating the complete destruction of an electron and a positron by making them annihilate into gamma rays (a type of photon).
A: You have a box of pebbles. There are some pebbles that do not move at constant velocity when placed closed to one another. We repeat the experiment many times with these pebbles and characterize their motion. We make a plot of their acceleration as a function of their position and we notice that the acceleration is always proportional to the inverse distance squared between the two pebbles. That is,
$$a(x) = \frac{A}{r^2}$$
You take one of these objects and make it your very special standard. You paint it red and write "STANDARD" on it.
As you have a big box of these objects, you start playing with them and your standard. You notice that while the acceleration of these pebbles is always proportional to the inverse distance squared, the proportionality constant differs from pair to pair of pebbles. For one, the acceleration of the two pebbles, while always in opposite directions, are not the same. You define a value $M$ for each object, which is the ratio of the acceleration of your standard to the absolute value of their acceleration. That is, if the proportionality constants turn out to be $A_\text{standard}$ and $A_x$ for some pebble called "X", you define $M_x = \dfrac{|A_x|}{|A_\text{standard}|}$.
Great! So you go to write down $A_\text{standard}$ once and for all. But you notice that for each pair, the proportionality constant is different. So what do you do? You dig and dig into the box until you find another one of these objects that acts exactly like your standard. You can substitute it for your standard and you'll always get the same result. You also label it "STANDARD." Now you take these two, and measure the proportionality constant of their acceleration, and you call it $A$. If you use one standard pebble and one other pebble "X," the proportionality constant of the acceleration of the standard pebble will not be $A$, but something else, call it $A^\prime$. So we make up another number $Q = A^\prime/A$, which if you divide the acceleration of the standard pebble, you get back $A$.
This whole pebble business is silly, but that's kind of the point. Pairs of point charges repel or attract. This is characterized by the acceleration felt by both points. In general the accelerations could be anything. So we need two parameters to characterize them. One is the mass ratio of the two points, which determines the "asymmetry" between the two accelerations, or their ratio. Once the ratio between the two accelerations are fixed, all you need to characterize them is the absolute value of one of them, which is the charge.
A: Definitions in physics are always somewhat circular, because they are not really definitions. Instead, they're descriptions of the world. The way to make sense of "circular definitions" is usually to think about the experiments that led to those definitions.
Suppose you're Charles-Augustin de Coulomb. One day you discover that by rubbing objects on each other you can leave them in a state such that there is a force between them. Experimentation shows that this force goes inversely as the square of the distance (as long as the objects aren't moving too quickly, but Coulomb didn't know this). You can also see the following: Suppose you have objects $A$, $B$, $C$. You find that there is an attractive force $F_A$ between $A$ and $C$, and there is also an attractive force $F_B$ between $B$ and $C$. You discover that if you put $A$ and $B$ together, the force between $AB$ and $C$ is $F_A + F_B$.
This last statement is very important, because it means that the force is proportional to some property of these objects. You can call that property "charge" (numerically defining charge is just a matter of choosing a unit charge). Now you define the electric field $\mathbf{E}$ as the force per unit charge: that is, $\mathbf{E} = \lim_{q\to 0} \mathbf{F}/q$, where $\mathbf{F}$ is the force felt by the charge $q$.
You should also keep in mind that the statements you quote aren't even meant to be definitions; they're just rough descriptions of what charge and EM fields are.
I recommend that you take a look at chapter 12 of the Feynman Lectures (Vol 1). It deals with a similar issue regarding Newton's second law: Is $F=ma$ a definition of force?
