Let me add to Ivan's answer another equivalent
characterization of weak compactness, which might appeal to
you, as it makes them resemble miniature measurable
cardinals.

Namely, if $\kappa$ is a cardinal and
$\kappa^{&lt;\kappa}=\kappa$, then $\kappa$ is weakly compact
if and only if for every transitive set $M$ of size
$\kappa$ with $\kappa\in M$, there is another transitive
set $N$ and an elementary embedding $j:M\to N$ having
critical point $\kappa$.

This embedding characterization admits myriad forms. For
example, one can insist that $M\models ZF^-$ or even
$M\models ZFC$, and that $M^{&lt;\kappa}\subset M$, or that
every $A\subset \kappa$ can be placed into such an $M$, and so
on. One can even insist that $j\in N$, a property known as
the Hauser property.

These various embedding formulations of weak compactness
allow one to borrow many of the methods and techniques from
much larger large cardinals, which are most often described
in terms of embeddings, and apply them with weakly compact
cardinals. For example, using Easton support forcing
iterations, one can control the value of $2^\kappa$ while
preserving the weak compactness of $\kappa$.

On the definition of weakly compact cardinals
What is the difference between them in terms of meaning and usage in daily life.

Can we say :


  I have a Intuition  / Premonition / Precognition / Instinct /  Gut
  feeling / Hunch you are going to pass
  driver test successfully.
  
  I have a  Intuition  / Premonition / Precognition / Instinct /  Gut
  feeling / Hunch that if I accept this job offer, it is not going to affect
  my career positively.
  
  I have a Intuition  / Premonition / Precognition / Instinct /  Gut
   feeling / Hunch that my son and your son are going to get along well even they have not met yet.

