Firstly, how is $NaOH$ created? My teacher said that before it is $NaOH$, it is $NaO^-$.
$$NaO^- + H_2O = NaOH + OH^-$$
If so, that means that $NaO^-$ is a base, which would make $NaOH$ it's conjugate acid. However, $NaOH$ is a base. So, perhaps my teacher said wrong, and the fact is, the "prelude" to $NaOH$ is $NaOH_2^+$, i.e.:
$$NaOH_2^+ + H_2O = NaOH + H_3O^+$$
Meaning that $NaOH_2^+$ is the acid, and $NaOH$ is the conjugate base.
Okay, so assuming the latter is correct, I still don't see how NaOH is a base.
Bases, in their most general definition, are substances that easily accept proton(s). Now, this would make sense for $NaOH$ if the equation looked like this:
$NaOH + H_2O = NaOH_2^+ + OH^-$
This would make the solution more basic, due to the higher concentration of $OH-$, though considering how strong of a base $NaOH$ is, that means that $NaOH_2^+$ is a weak acid, and the solution would therefore revert back a lot (double arrow thingy).
However, I have never seen the equation like that. Instead, I've seen it like this:
$$NaOH + H_2O = Na^+ (aq) + OH^-(aq)$$
Now, that would increase the concentration of $OH-$, would would make the solution more basic, but there is no proton being accepted here. The so-called base that is $NaOH$ does not accept a proton, but rather releases an $OH^-$ group, which is quite the opposite of accepting a proton. Now, this would make sense if $NaOH$ was classified as "a basic salt", as in, a salt that makes a solution more basic. So, with such an explanation, $NaOH$ is just a salt, not a base, but it is a salt that makes the solution more basic.
Now, since it is a substance that makes a water solution more basic, it would by definition become an Arrhenius base. I can accept that, because the Arrhenius base definition only requires the substance to make the water solution more basic. $NaOH$ does that. However, I cannot see in any way how $NaOH$ fits the Brønsted-Lowry definiton of a base.
Please explain to me simply, but thoroughly, if you can. For some reason, the Acid-Base part of chemistry has evaded my comprehension for a while now.