If an Ideal Current Source has infinite resistance, how is current flow possible? Where does current come from if an ideal current source has infinite resistance? Current can't move through the current source because of the resistance, so I don't really understand how and why an ideal current source has infinite resistance.
 A: I think you can answer your question by contrasting the simplest possible circuit simply by connecting your source to an external resistor R.  For the ideal voltage source, which is essentially a battery, you want negligible resistance inside the source, so the total resistance is very close to R, and then you'll get I=V/R flowing through it, so I will depend on R but V won't (whereas if there was resistance R' within the source, you'd get a reduction in the effective V by IR', so that would be close to VR'/R, and you'd only have 1-R'/R of the V you want).  But notice the key point-- if you vary R, what stays fixed is V, not I, so you have a good V source but a lousy I source.
To get a good I source, go to the opposite limit of a huge R' in the source.  Then it won't matter the R you attach to the source, because the R' will dominate anyway.  You'll always get close to I=V'/R', for any R, so I stays close to constant since V' does.  However, the V over the resistor R will obey V=IR=V'R/R', so it won't be constant at all, it will be proportional to R.  
Your question is, in the limit as R' goes to infinity, doesn't this kill the I, but the answer to that is, if you want a fixed I source, you need to ramp up V' in proportion to R'.  Neither R' nor V' can really be infinite, but the trick is you must make them both very large (and proportionally so) to get a fixed I source.  I suppose this makes the answer to your question, "that's why it is impossible to have a truly ideal current source."
A: An ideal current source with infinite internal resistance is an abstraction used in circuit theory. As the name says, it is just the source of a constant current that does not react to an applied voltage. Therefore the infinite resistance. A real current source is modeled by an ideal current source yielding a constant current in parallel with a finite internal resistance.  
A: 
Where does current come from if an ideal current source has infinite
  resistance?

The slope of the I-V curve for an ideal current source is zero just as in the case of an infinite resistance.  To say that an ideal current source has infinite internal resistance is to say that the current through is independent of the voltage across.
From another perspective, for a current source with finite internal resistance $r_s$, the load resistance is in parallel with the internal resistance and so, by current division, the load current varies with the load resistance and is given by
$$I_L = I_S \frac{r_s}{r_s + R_L}$$
See that, in the limit as $r_s \rightarrow \infty$, the load current becomes the constant $I_L = I_S$.  Thus, we say that an ideal current source, which produces a load current independent of $R_L$, the internal resistance is infinite.
A: "Ideal" in this context means that no consideration is being given to how the ideal properties (constant current regardless of load, infinite resistance) can be achieved in practice. As freecharly puts it, it is an abstract concept. 
The process of analysing electrical circuits starts with modelling real electrical components, which have several properties, as combinations of ideal elements (like those in chemistry) which each have a single property - pure resistor/capacitor/inductor, pure voltage/current source, etc. 
No real current source has the properties of an ideal current source, and it is as pointless to ask how an ideal current source can have infinite resistance yet generate a finite current, as it is to ask how any real string can be inextensible yet exert a finite tension when you pull on it. However, for the purpose of analysing electrical circuits most real current sources can be modelled with sufficient accuracy as an ideal current source in parallel with a finite internal resistance. A real current source will also have some capacitance and inductance, which become significant at high frequency.
