What exactly are induction and air ionization (static electricity) ?

To understand induction, you have to first understand the
basic structure of all matter.  Remember that all matter is composed of atoms which have
a positive nucleus surrounded by negative electrons.  The nucleus is fixed in place but
the electrons have varying degrees of ability to move within a substance.  In metals,
the electrons are considered to be very mobile and are referred to as an "electron
cloud" because of how they can shift within the metal.  Second you have to take into
account that opposite charges attract while like charges
repel.

An electroscope detects charge - either positive or
negative - because of these two factors:  electron mobility and charge
interaction.

Lets consider what happens when a positively
charged object is brought near an electroscope.  Because the object has a positive
charge, the electrons in the metal leaves of the electroscope will be attracted toward
the top of the electroscope.  When some of the electrons move toward the top, the result
is that the leaves now acquire a temporary negative charge.  Because both leaves have a
positive charge, they move away from each other.  The separation of the leaves indicates
the object being tested has a charge.

Likewise, if a
negatively charged object is brought near the top of the electroscope, electrons in the
metal rod will be repelled and move toward the leaves at the bottom producing a
temporary negative charge in the leaves.  Because both leaves have the same charge the
leaves again move apart.

Induction is a temporary process.
As soon as the charged object is removed the electrons in the electroscope return to
their original position and the electroscope is again
neutral.

Static electric charges result when electrons move
from one object to another as a result of actual contact between the two objects. When
the objects collide some electrons can be physically transferred from one object to
another.  This is what causes the build up of charge within storm clouds, ultimately
resulting in a discharge of lightning as the separated charges
recombine.