What is actually a redox reaction?
A redox reaction is any reaction in which oxidation and reduction are occurring. The word redox is a portmanteau of REDuction-OXidation. So, if we can correctly identify reduction and oxidation, we can identify redox reactants. In practice, this can be a little challenging, so I will focus first on the second part of your questions (definitions) and then go back to the reaction.


  Ionization is any process that produces ions. 


The conversion of a hydrogen atom to a hydrogen ion by loss of an electron is an ionization. 
$$\ce{H -&gt; H+ + e-}$$
So to is the reaction of hydrochloric acid and water, which produces hydronium ions and chloride ions.
$$\ce{HCl + H2O -&gt; Cl- + H3O+}$$


  Oxidation is a loss of electrons.


The ionization of hydrogen atoms to hydrogen ions is an oxidation.
$$\ce{H -&gt; H+ + e-}$$

So to is the conversion of carbon monoxide to carbon dioxide:

$$\ce{2CO + O2 -&gt; 2CO2}$$

Note that this type of reaction is what gave us the word oxidation - compounds are reacting with oxygen and atoms end up with more oxygen atoms bound to them. 


  Reduction is a gain of electrons.


The ionization of chlorine atoms to chloride ions is a reduction.

$$\ce{Cl + e- -&gt; Cl-}$$

So too is the conversions of acetylene to ethane:

$$\ce{C2H2 + 2H2 -&gt; C2H6}$$

Not all oxidations and reductions are ionizations. There is a simple mnemonic in English for remembering oxidation and reduction: OIL RIG (Oxidation Is Loss Reduction Is Gain).


  Oxidation and reduction happen together. 


In principle we can write an equation that shows only oxidation:

$$\ce{H -&gt; H+ + e-}$$

but oxidation cannot happen on its own. That electron that was lost needs to be picked up by something else. This reaction that shows only oxidation is called a "half-reaction" and needs to be coupled with a reduction half-reaction to show a valid chemical reaction:

$$\ce{H -&gt; H+ + e-}$$
$$\ce{Cl + e- -&gt; Cl-}$$
$$\ce{H + Cl -&gt;HCl}$$

In the other examples I gave, both processes are occurring. In the oxidation of carbon monoxide, the oxygen molecules are being reduced.

The surest way to identify oxidation and reduction (and thus redox) is to compare the oxidation states of the reactants and the products. Determining oxidation states/numbers is more complex that I want to put here, so start with the Wikipedia article.

For the carbon monoxide example:

                  reactants    products
C                 +2           +4
O (bonded to C)   -2           -2
O2                0            -2


The carbon atoms are being oxidized (oxidation number increased) and the oxygen atoms in $\ce{O2}$ are being reduced (oxidation number decreases). The oxygen atom already attached to carbon did not change its oxidation number.