Chocobos are sorted into two Classes:
Mental     and    Physical.
 
It's important to read about and learn the difference between these two categories, as it directly affects which type of Careplan you
put your Chocobo on, even from the very start.

The two Stats that directly coincide with Mental Chocobos are:
Discernment & Receptivity.

Discernment:  Affects the chocobo’s potential
to learn abilities.  Basically, the higher your Chocobo's Discernment, the more likely your Chocobo will learn an ability from a Story.
Keep in mind each story requires it's own particular amount of Discernment to learn that individual story.  For Instance, Canter requires a much higher amount of Discernment than Auto-Regen.  Your Chocobo's mood and age also affects the likelihood of learning an ability.

In Chocobo Racing, increased Discernment affects your chocobo's pace settings and its item-use timing as well.

Another thing I'd like to note is many people have claimed to notice that high Discernment greatly increases rewards and findings while Chocobo Digging.  Although there seems to be no direct evidence, there have been numerous reports of this corelation.

Discernment is also the Mental Stat which affects the aesthetics of your Chocobo.  Religiously following a Mental Program from the start should grant you Average Discernment+ by day 30.
Acquiring average Discernment by day 29 (Adulthood) grants your Chocobo a particular appearance:  Raptor Beak and long head feathers.  Note:  If your Chocobo does not attain average Discernment by day 29, you will not acquire either of these physical traits.

Discernment and Receptivity are boosted by feeding your Chocobo Zegham Carrots and
Azouph Greens.  During the Chick Phase, Herb Pastes can be fed to your Chocobo to slightly affect Discernment.  Tokopekko Wildgrass is a medicinal herb that slightly increases Discernment as well as keeps your Chocobo Healthy.
Reading Stories will noticeably boost Discernment.  Mental Careplans like Listening to Music during Adolescence and Exhibiting to the Public later on increase both Discernment and Receptivity.
Particular Careplans like Playing with Children and Digging for Treasure later on Isolate and Boost just Discernment.

Receptivity: Affects the chocobo’s ability to communicate with fellow chocobos.
Having Receptivity as a chocobo's dominant stat results in a sensitive personality.

Receptivity increases the rate at which you acquire new stories.  The higher your Chocobo's Receptivity, the more frequent NPCs appear in walking Cut Scenes.

Receptivity can be isolated and boosted by setting your Chocobo on Careplans such as Playing with chocobos and Acting in a play
later on.

Receptivity also seems to increase a bit faster than Discernment and should hit Average a few days earlier than Discernment hits Average.

It also seems to me that a parent card with higher Receptivity increases the odds that an offspring will bear one of their abilities.  It just seems a logical assumption but there is no direct evidence of this.

In Chocobo Racing, increased Receptivity affects the Chocobo's ability to avoid accidents.

Mental Chocobos usually end up being either Diggers or used as Breeding Holders
with ideal Physical abilities for future Breeders.

Diggers:
The reason why Mental Chocobos are diggers
is because you need a very high amount of Discernment to learn Burrow and Bore, which are the two best Digging Abilities. 
Click here to read up on Chocobo Abilities.
Both Burrow and Bore can usually be learned with Outstanding Stats, but under the right circumstances, may also be learned with Impressive Stats.

Increasing your Chocobo's Endurance after acquiring both Digging Abilities, will allow your Chocobo to stay out for longer while Digging.

A lot of Breeders raising Diggers also try to
look for parental cards bearing either Burrow or Bore, since it's time consuming acquiring both abilities and takes a lot of Cut Scenes meeting NPCs.

Breeding Holders Chocobos:
Many Chocobo Breeders will raise Mental Chocobos with Max Discernment and Receptivity along with two Physical Abilities
like Canter and Gallop strictly for future Breeding purposes.

This is obviously not beneficial for the Breeder if they are using their Chocobo for Racing since their Chocobo will end up being dead-slow.  However it's lucrative if they want to then breed an offspring inheriting one of these two abilities,
or sell this parental card bearing two desirable abilities.  In addition, the Chocobo's high Receptivity increases odds that the offspring
will inherit one of the two abilities.

 
Some Raisers follow a Mental Program until both abilities are acquired, either inherited + learnt, or both learnt, then switch to a Physical Program to bring up Physical Stats.
To learn in-depth about this Multi-Step Physical + Mental Program, read the Careplan carefully.