To the question "what is Impedance," I would note that impedance is a broad concept of physics in general, of which electrical impedance is only one example.
To get a grasp of what it means and how it works, it's often easier to consider mechanical impedance instead. Think of trying to push (slide) a heavy couch across the floor.
You apply a certain amount of force, and the couch slides at a certain velocity, depending on how hard you push, the weight of the couch, the type of floor surface, the type of feet that the couch has, and so on. For this situation, it's possible to define a mechanical impedance that gives the ratio between how hard you push and how fast the couch goes.
This is actually a lot like a dc electrical circuit, where you apply a certain amount of voltage across a circuit, and current flows at a certain corresponding rate through it.
For the case of both the couch and the circuit, the response to your input may be simple and fairly linear: a resistor that obey's Ohm's Law, where its electrical impedance is just the resistance, and the couch may have friction slider feet that allow it to move with a velocity proportional to your force.*
Circuits and mechanical systems may also be nonlinear. If your circuit consists of a variable voltage placed across a resistor in series with a diode, the current will be near zero until you exceed the forward voltage of the diode, at which point current will begin to flow through the resistor, in accordance with Ohm's law. Likewise, a couch sitting on the floor will usually have some degree of static friction: it won't begin moving until you push with a certain amount of initial force. In neither the mechanical nor electrical system is there a single linear impedance that can be defined. Rather, the best that you can do is to separately define impedances under different conditions. (The real world is much more like this.)
Even when things are very clear and linear, it's important to note that impedance just describes a ratio-- it doesn't describe the limits to the system, and it's not "bad." You can definitely get as much current/velocity as you want (in an ideal system) by adding more voltage/pushing harder.
Mechanical systems also can give a pretty good feel for ac impedance. Imagine that you're riding a bicycle. With each half-cycle of the pedals, you push left, push right. You can also imagine pedaling with just one foot and a toe-clip, such that you push and pull with every cycle of your pedal. This is a lot like applying an ac voltage to a circuit: you push and pull in turn, cyclicly, at some given frequency.
If the frequency is slow enough-- like when you're stopped on the bike, the problem of pushing down on the pedals is just a "dc" problem, like pushing the couch. When you speed up, though, things can act differently.
Now, suppose that you're biking along at a certain speed, and your bike is a three-speed with low, medium, and hi gear ratios. Medium feels natural, hi gear is difficult to apply enough force to make any difference, and at low gear, you just spin the pedals without transferring any energy to the wheels. This is a matter of impedance matching, where you can only effectively transfer power to the wheels when they present a certain amount of physical resistance to your foot-- not too much, not too little. The corresponding electrical phenomenon is very common as well; you need impedance matched lines to transmit RF power effectively from point A to point B, and any time that you connect two transmission lines together, there will be some loss at the interface.
The resistance that the pedals provide to your feet is proportional to how hard you press, which relates it most closely to a simple resistance-- particularly at low speeds. Even in AC circuits, a resistor behaves like a resistor (up to a certain point).
However, unlike a resistor, the impedance of a bicycle is dependent on frequency. Suppose that you put your bike in high gear, starting from a stop. It can be very hard to get started. But, once you do get started, the impedance presented by the pedals goes down as you get going faster, and once you're going very fast, you may find that the pedals present too little impedance to absorb power from your feet. So there's actually a frequency-dependent impedance (a reactance) that starts out high and gets lower as you head to higher frequency.
This is much like the behavior of a capacitor, and a fairly good model for the mechanical impedance of a bicycle would be a resistor in parallel with a capacitor.
At dc (zero velocity), you just see the high, constant resistance as your impedance. As the pedaling frequency increases, the capacitor impedance becomes lower than that of the resistor, and allows current to flow that way.
There are, of course, various other electrical components and their mechanical analogies**, but this discussion should give you some initial intuition on the general concept to stay grounded (pun intended) as you learn about the mathematical aspects of what can at times seem like a very abstract subject.
*A word to the picky: Ohm's law is never exact for a real device, and real-world friction forces never give velocity exactly proportional to force. However, "fairly linear" is easy. I'm trying to be all educational and stuff here. Cut me some slack.
**For example, an inductor is something like a spring-loaded roller on your wheel that adds drag as you get to higher frequency)