Triviality of $H_3(G,\mathbb{Z})$ We know that the triviality of the Schur multiplier means that projective representations can be lifted to ordinary ones. The Schur multiplier is also a measure of the failure of how the commutator identities fail to follow from the universal ones.
My Question: What does the triviality of $H_3(G,\mathbb{Z})$ mean?
 A: $H_3(G, \mathbb{Z})$ is (noncanonically) isomorphic to $H^3(G, \mathbb{C}^{\times})$. Let me instead answer the following more general question: what does it mean for $H^n(G, A)$ to be trivial, where $G$ acts on $A$ trivially? In the treatments that I've seen this question is only answered for $n = 1, 2$, leaving the higher cohomology something of a mystery. 
Recall that $H^2(G, A)$ classifies central extensions of $G$ by $A$, which I'll write as short exact sequences
$$1 \to A \to E \to G \to 1.$$
There is an analogous statement for $H^n(G, A)$, but my favorite version of this statement requires talking, not about groups, but about spaces. Namely, we can think of the above short exact sequence as a fiber sequence of spaces
$$BA \to BE \to BG$$
where $BG$ denotes the classifying space of a group, or equivalently in this case the Eilenberg-MacLane space $K(G, 1)$. This is the space whose homology is group homology and whose cohomology is group cohomology. $H^2(G, A)$ is the set of homotopy classes of maps $BG \to B^2 A$, where $B^2 A$ is another way of writing the Eilenberg-MacLane space $K(A, 2)$, and the relationship between this description of group cohomology and the above description is that $B^2 A$ is the "classifying space of principal $BA$-bundles," and a principal $BA$-bundle over $BG$ turns out to be equivalent to a central extension of $G$ by $A$; starting from a map $BG \to B^2 A$, you get the maps above by taking homotopy fibers twice.
Now, that was a lot of homotopy theory just to restate a fact about group cohomology that you already knew, but the point is that homotopy theory tells you what higher group cohomology means too. Namely, starting from a map $BG \to B^{n+1} A$, taking homotopy fibers twice gives a fiber sequence
$$B^n A \to X \to BG$$
where $X$ is a space with $\pi_1(X) \cong G, \pi_n(X) \cong A$, and all other homotopy groups trivial. You can think of such a space as a higher categorical generalization of a group called an "$n$-group," and now the point is that $H^{n+1}(G, A)$ classifies a particular kind of such spaces (the ones where the natural action of $\pi_1$ on $\pi_n$ is trivial). This is the simplest nontrivial special case of the general theory of Postnikov towers, and is for my money the correct generalization to higher cohomology of the familiar fact about $H^2(G, A)$ classifying central extensions. 

Okay, but that was a lot of abstract stuff, probably too much if you haven't seen a decent amount of algebraic topology before. Let me tell you a relatively concrete model of this in the special case of $H^3(G, \mathbb{C}^{\times})$. 
First, here's a fun interpretation of $H^2(G, \mathbb{C}^{\times})$. Start with the group algebra $\mathbb{C}[G]$. Think about twisting the multiplication by a constant: that is, think about functions $c(g, h) : G \times G \to \mathbb{C}^{\times}$ such that if we define
$$g \circ h = c(g, h) g h$$
then $\circ$ makes $\mathbb{C}[G]$ into an algebra. It turns out that $c(g, h)$ must be a $2$-cocycle, and moreover that if $c, c'$ are cohomologous $2$-cocycles then the corresponding algebras are isomorphic. These are what might be called "twisted group algebras" of $G$, and their representation theory describes projective representations of $G$ twisted by the corresponding $2$-cocycle. So far this is just a repackaging of stuff you already know.
Now let's go one category level up. Instead of thinking about the algebra $\mathbb{C}[G]$ let's think about the monoidal category $\text{Vect}[G]$ of $G$-graded vector spaces. This is the category whose objects can be thought of as formal direct sums $\bigoplus_{g \in G} V_g g$ of vector spaces indexed by elements of $G$ (analogous to how $\mathbb{C}[G]$ consists of formal sums of complex numbers indexed by elements of $G$) tensoring in the obvious way suggested by the notation, namely
$$\left( \bigoplus_{g \in G} V_g g \right) \otimes \left( \bigoplus_{g \in G} W_g g \right) \cong \bigoplus_{g \in G} \left( \bigoplus_{h_1 h_2 = g} V_{h_1} \otimes V_{h_2} \right) g.$$
In the same way that $\mathbb{C}[G]$ controls how $G$ acts on vector spaces, $\text{Vect}[G]$ controls how $G$ acts on certain categories. And there's a question analogous to the question of how to twist the multiplication here, namely how to twist the associator isomorphism
$$(V_{g_1} \otimes V_{g_2}) \otimes V_{g_3} \cong V_{g_1} \otimes (V_{g_2} \otimes V_{g_3}).$$
We can try to multiply the associator isomorphism by a complex number $c(g_1, g_2, g_3) \in \mathbb{C}^{\times}$, and then it turns out that the axioms of a monoidal category imply that we get another associator iff $c$ is a $3$-cocycle, and moreover cohomologous $3$-cocycles give equivalent monoidal categories. In other words,

In the same way that the second cohomology $H^2(G, \mathbb{C}^{\times})$ describes how we can twist the multiplication on $\mathbb{C}[G]$, the third cohomology $H^3(G, \mathbb{C}^{\times})$ describes how we can twist the associator on $\text{Vect}[G]$.

In particular, $H^3(G, \mathbb{C}^{\times})$ vanishing ought to tell you something about "projective representations of $G$ on categories," although I'm not sure how to make this precise. 
The relationship between this story and the homotopy theory story comes from thinking about $\text{Vect}[G]$, twisted by a $3$-cocycle, as a $2$-group. But this answer is already a bit too long so maybe I'll stop here. Let me just mention that one reason to care about these twisted categories is that they describe a topological quantum field theory called Dijkgraaf-Witten theory in $3$ dimensions: in particular, each one gives rise to $3$-manifold invariants and representations of mapping class groups of surfaces. 
