How is an electric field with a curl possible? Is there any example for curl of electric field? According to Maxwell's 3rd equation there should be curl of electric field for changing magnetic field.
Are there any real time examples where curl or rotation of electric field is possible?
 A: Sure! Let's take the example which led Einstein to pursue his special theory of relativity. 
Suppose you take a loop made of wire and put it on your desk, so it is horizontally-flat. You now put the N pole of a magnet underneath some part of your desk, so that there is a (spatially limited!) magnetic field going "up" through some part of your desk, perpendicular through it. Since the magnetic field is constant, and we're not using batteries or anything, there is no appreciable electric field here.
Now we know that when we pass this wire loop over the magnet we see an AC current. Let's see how that can come about without electric fields. Let's imagine that we're looking from behind the loop in the direction it's travelling, so that you know what I mean when I say "forwards", "left", "right", "up", and "down".  We know that when the loop moves forwards through an upwards magnetic field, all of the electrons want to curve left, and all of the protons want to curve right. These are balanced in our loop, so there is no net force, but there can be a net current, if either the protons start rotating (which requires the loop to rotate, which is hard because it's big and heavy) or if the electrons start rotating (which is just an internal electric current, that's pretty easy). When does this happen?
Well, while the loop enters this magnetic field, the field is stronger at the front of the loop than it is at the back. This leads to a stronger left-curvature of the electrons at the front which wins out (the back ones are not curving left yet!) and pushes the electrons at the back "out of their way" to the right, which create a current where the electrons flow counterclockwise around the loop when seen from the top. (We'd call this a clockwise current since electrons have negative charge.) Then if the loop is a lot smaller than the magnet you'd see the loop enters the magnetic field completely, a region of constant magnetic field, and the tendencies of the electrons to curve left evens out between the front and the back: no more current flows. Finally as it exits the magnetic field, the front-most electrons no longer want to curve so much left, but the back-most ones still do, leading to an opposite current. You can use physics to calculate the exact magnitude of the current, if you like; it's a standard undergraduate exercise.
Here is another way to calculate the exact same quantity: fix the square in place on the desk, and move the magnet under it, with the same speed as the loop had!  In this perspective, the magnetic field at each point starts changing, first increasing as the magnet moves closer under it, and then decreasing as the magnet moves further. According to this equation the electric field curls first clockwise and then counterclockwise. And you can work out how much current this curling electric field generates in the loop, and it works out to be the exact same quantity. (In fact the magnetic fields cannot at first create a force on the electrons in this reference frame, because in this reference frame the electrons are standing still!)
Einstein was like, "That can't be a coincidence. Clearly this rule of 'it doesn't matter whether I move over the magnet or the magnet moves under me' is a bigger law of physics." And this gave him a faith in the laws of electrodynamics which his contemporaries didn't have -- they thought "well we're getting some weird predictions from these laws, clearly they're wrong." But he was like, "these laws point at this deeper law, maybe they're 100% correct and the assumptions we're bringing into Newton's laws are wrong." He came up with his theory of special relativity mostly by swiping the cutting-edge math of the electrodynamics of his day, and arguing that you had to actually take it seriously, and showing that since it's mathematically consistent it cannot, ipso facto, create contradictions.
So, see, whenever magnets are moving through space we have to describe them as changing the electric fields as well as the magnetic field -- otherwise our mathematics could not possibly be consistent with the principle that physics works the same on a train or in a car (moving at constant velocity) as it does on the ground. It's as simple as asking "what currents are created in a wire loop laying on the train when it passes over a big magnet on the ground?" -- there are two ways to look at the electromagnetic field, and in one it's all purely magnetic, so in the other one it must be partly electrical and partly magnetic.
