Questions on Inductor Selection guide from TI
Does it mean that this report applies to buck converter which always
  work in Continuous Conduction Mode?


The mode a buck converter operates in is determined by the conditions it is operating in, not the converter, and any buck converter can be forced into both modes.  There is no such thing as a buck converter which only works in one mode.  

However, it is desirable to design a converter such that it remains in only one conduction mode.  So this app note isn't so much something applicable to a specific conduction mode, but rather how to chose an inductor that will permit a given buck converter to stay in continuous conduction mode.  Ultimately, the higher the inductance, the lower the minimum load can be before the converter is forced to transition to discontinuous conduction mode (DCCM).  

Buck converters that are constant frequency and current-mode are almost always optimized for continuous conduction mode (CCM), but there are voltage-mode variable frequency/PFM (pulse frequency modulation) converters optimized for DCM as well. 


  What will happen when converter switch to discontinous conduction mode
  on small load?


It depends on the converter.  Generally, at least for ones optimized for CCM (most of them), there will be a sudden increase in output voltage ripple, which in turn results in additional losses in the output capacitor and reduced efficiency.  It is also not uncommon for additional ringing to occur, causing more EMI.  Things like voltage regulation and transient response will also usually suffer somewhat, and in some cases, certain buck regulators may not be able to regulate at all.  These are uncommon however, and will have a very clear 'minimum load' spec in the datasheet.  Also note that this is not dissimilar from most linear regulators, which will also go out of regulation if the load drops too much.  

More advanced converters will often implement a completely different control loop that will be switched to for loads too small to maintain CCM.  Often it will even change the switching behavior, but not necessarily.  You may see terms like burst mode or pulse skipping mode, which different names for the same thing - it is a voltage-mode PWM control loop that can skip entire pulses.  This is better suited to DCM and can improve efficiency at light loads.  However, it is still fixed frequency so the other downsides are mostly still present (more ripple, poorer regulation, etc).  

Generally, you should try to keep a buck converter in the mode it is optimized for, but read the datasheet, it usually isn't the end of the world if it is required to go into DCM.  

It is also worth mentioning a third option.  Some converters have what is called forced continuous conduction mode.  This requires a synchronous converter (uses mosfets for both switches, instead of a mosfet and a diode), but instead of turning off for part of a switch cycle like it would in DCM, it crosses 0 current, but lets it continue in the opposite, negative direction, forcing continuous conduction.  This costs you some efficiency, but allows the same control loop to operate and doesn't result in increased ripple or poorer regulation.  

So, it's a trade off.  

Now, on to...

Inductor selection

Selecting an inductor can be as hard or as easy as you need or want it to be.  For any applications, you can worry about volt seconds, Bmax values, amp turns, and a good deal of things not even touched upon in the TI app note (mostly regarding core material and behavior).  But if you need to depends on the design requirements, and I would argue that for the vast majority of designs, you can completely ignore all of that and just find something that will not saturate and not get too hot while having high enough inductance to ensure at your desired load ranges, you can stay in your preferred conduction mode.  

However, this will usually only let you narrow things down to a large range of options that will work, and to a lesser extent, work well.  But if you have very challenging thermal, efficiency, or  size requirements (or some combination thereof), then you may need to do a more detailed evaluation of suitable inductors.  

This is where things get tricky:  different inductors will have magnetic cores made of different 'genres' of materials, and worse, each genre can have multiple types.  Each one will have losses and heat up that are independent and superimposed on the resistive losses in the winding.  

Ferrite, for example, has a non-linear loss relationship with the Bmax, or maximum field strength.  So it is easy to imagine situations where in addition to simply ensuring the current rating and inductance is sufficient, you might want to also make sure the B field in the magnetic core would be beneath a certain value to optimize core losses.  

Furthermore, frequency impacts this as well.  

The inductance generally will remain more or less the same, with changing frequency, but losses in the core will not.  And this is where things get really complicated, because this is highly dependent on the actual core material used.  

You will find two main classes of inductors (again, this is a very broad generalization, this is not some hard or fast rule, but simply a loose pattern you might find on Digikey's inductor selection).  Ferrite, and carbonyl iron.  Ferrite is a magnetic ceramic and is only slightly conductive (tens to hundreds of kΩ across common core cross sections and distances).  This means losses caused by induced eddy currents in the core are very low.  

The darker side to this is ferrites also have a very strong negative resistance coefficient with temperature.  So it will become more conductive as it heats up, which will cause eddy current losses in the core to go up meaningfully as the core gets hotter.

Ferrites also saturate like a brick wall.  If you saturate a ferrite inductor in any sort of DC/DC application, you can expect permanent damage to the circuit to result.  

Ferrites are also not very good at storing energy.  If you need a lot of inductance, ferrite cores will be physically large at any non-trivial current.  The 'power' ferrite materials usually saturate at ~300mT, so that limits how much inductance you can achieve at a given saturation current.  

Anyway, common ferrite inductors will not see significant increases in losses below 400-500kHz.  500kHz is the rule of thumb for the maximum 'low end' ferrite frequency range.  There are other ferrite material designations that are lower loss at higher frequencies: 



Unfortunately, often you are at the mercy of the inductor datasheet.  My advice is that if you can't find the information you need in the datasheet for an inductor, then you probably shouldn't use that inductor in that application.  Leave it for applications where the information in the datasheet matches the information you even need to worry about.  

Carbonyl iron is a composite, also known as iron powder.  It comes in an even wider variety than ferrite, but generally it has a few common and interesting properties:  


Significantly higher eddy current losses compared to ferrite.  Carbonyl iron is hydrogen reduced iron, a fine powder of iron with insulating impurities mixed in, but it is a good deal more conductive than its ceramic ferrite cousins. 
Linear Saturation.  The inductance decreases linearly with increasing current above a certain point.  
Lower loss at very high frequencies (for switchers, 2-3MHz+) than ferrites, due to their extremely low hysteresis loss.  And this is despite their higher eddy current loss.  
Much higher energy storage.  A carbonyl iron core can yield much more inductance at much higher currents while being significantly smaller than ferrite cores.  And core loss is volumetric - less material means less loss.  


It is one very complex, multifaceted trade off and there are all sorts of circumstances and requirements where one type of core would actually be better than the other, sometimes unintuitively so, once you factor in all the different core loss factors.  

But, in general, for anything below 500kHz, don't even worry about frequency.  At 1-2MHz, I personally start to favor iron powder and get good 'rough draft' results, but this is in situations where I don't need to worry that much about it.  

I've already gone well beyond the scope of this question I think, but my advice is to not worry about things until you actually need to worry about them.  And you should be fine simply going by the more common values you find in most inductor datasheets.  The reason that is what is used in most datasheets is because that is all most engineers use or need, most of the time.  

Higher performance will demand more math and footwork though. 

Final note on ESR:

Things like output capacitance and the ESR is totally controller specific.  If a controller/converter doesn't mention any specific ESR requirements, then it probably doesn't have any.  Different converters have different control loops compensated differently, but there is nothing unique or specific to buck converters regarding this loop or it's stability.  Just follow what the datasheet says.  

However, the output capacitors in buck converters are not all that important beyond ensuring loop stability and being able to handle the inductor ripple current.  The capacitors you should be worried about are the input capacitors - at least for buck converters.