What occurs when an iron nucleus accepts electrons?

Oxidation is formally the loss of electrons; whereas reduction is formally the gain of those electrons by another species.

You know the old doggerel: #"LEO says GER"#.

#"Loss Of Electrons, OXIDATION; Gain of Electrons, REDUCTION."#

In any chemical reaction, mass AND charge are conserved.

Here,

#Mg(s) + Fe^(2+)(aq) rarr Mg^(2+)(aq) +Fe(s)darr#

Are charge and mass conserved here? How do you know?

Chemists habitually break these reactions up into half-equations,

i.e. #"Reduction,"#

#Fe^(2+)(aq) +2e^(-) rarr Fe(s)#

#"Oxidation,"#

# Mg rarr Mg^(2+)(aq)+2e^(-)#

Electrons have been introduced into the equation as virtual particles, and when the oxidation and reduction are combined (here they are added 1:1), the electrons cancel:

#Fe^(2+)(aq) +2e^(-) + Mg rarr Mg^(2+)(aq)+Fe(s)#

Why do chemists do this? Well, (i) they wish to torment students, and (ii) for complicated redox reactions it is an efficient and systematic way of balancing equations.

For more examples of redox chemistry, see here and here.

To tell if something is reduced, look at the charge on chemical the left and right hand side. If the charge goes down (like Fe in your question, it goes from +2 on the left to 0 on the right), then it has been reduced, because it has gained electrons (electrons have a -1 charge). If the charge goes up, then it has lost electrons and has been oxidised.

A way to remember it is OILRIG, which stands for:

Oxidation
Is
Loss (of electrons, so the charge goes up in the reaction)
Reduction
Is
Gain (of electrons, so the charge goes down in the reaction)