If we have two groups $G,H$ the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product $G\times H$ into a group it is certainly by defining the multiplication
$$(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2),$$
with identity $(1,1)$ and inverse $(g,h)^{-1}=(g^{-1},h^{-1})$.
On the other hand we have the construction of the semidirect product which is as follows: consider $G$,$H$ groups and $\varphi : G\to \operatorname{Aut}(H)$ a homomorphism, we define the semidirect product group as the Cartesian product $G\times H$ together with the operation
$$(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1\varphi(g_1)(h_2)),$$
and we denote the resulting group as $G\ltimes H$.
We then show that this is a group and show many properties of it. My point here is the intuition.
This construction doesn't seem quite natural to make. There are many operations to turn the Cartesian product into a group. The one used when defining the direct product is the most natural. Now, why do we give special importance to this one?
What is the intuition behind this construction? What are we achieving here and why this particular way of making the Cartesian product into a group is important?
Best Answer
Forget about the actual construction of the semidirect product for now. I argue that the semidirect product is important because it arises naturally and beautifully in many areas of mathematics. I will list below many examples, and I urge you to find a few that interest you and look at them in detail.
Before doing that let me just give the following extra motivation: say you have a group $G$, and you find two subgroups $H,K$ such that every element of $G$ can be uniquely written as a product $hk$ for $h \in H$, $k \in K$. In other words, you have a set-theoretic bijection between $G$ and $H \times K$. Then certainly you'd want to understand $G$ by studying its smaller components $H$ and $K$. One way to achieve this would be to find a suitable group structure on $H \times K$ intertwining the structures of $H$ and $K$ such that the above bijection becomes a group isomorphism. This can be done; however doing things in this generality becomes rapidly tedious. If instead we restrict our attention to such decompositions with $H$ normal in $G$, the problem becomes much more manageable. In this case we have what we call a split exact sequence $$1 \to H \to G \to K \to 1,$$ and $G$ is called a semidirect product of $H$ and $K$. An existence and uniqueness theorem gives us all the possible semidirect products one can obtain from $H$ and $K$ through the group of homomorphisms from $K$ to $\operatorname{Aut}H$. Note that $K = \mathbb{Z}/2$ appears often in practice, because this guarantees normality of $H$. Now here are some examples:
The symmetric group $S_n = A_n \rtimes \mathbb{Z}/2$. The exact sequence is $$1 \to A_n \to S_n \xrightarrow{\mathit{sign}} \mathbb{Z}/2 \to 1.$$
The dihedral group $D_n = \mathbb{Z}/n \rtimes \mathbb{Z}/2$. The exact sequence is $$1 \to \mathbb{Z}/n \to D_n \xrightarrow{\mathit{det}} \mathbb{Z}/2 \to 1.$$
The infinite dihedral group $D_\infty = \mathbb{Z} \rtimes \mathbb{Z}/2$. The exact sequence depends on your explicit construction. You may take $$1 \to \mathbb{Z} \to \mathbb{Z}/2 * \mathbb{Z}/2 \to \mathbb{Z}/2 \to 1$$ or $$1 \to \mathbb{Z} \to A(1,\mathbb{Z}) \to \mathbb{Z}/2 \to 1,$$ where $A(1,\mathbb{Z})$ is the group of affine transformations of the form $x \mapsto ax + b$, where $a \in \{ \pm 1 \} \cong \mathbb{Z}/2$ and $b \in \mathbb{Z}$.
Many matrix groups, thanks to the determinant map. For example, $G = \operatorname{GL}(n,\mathbb{F})$, $O(n,\mathbb{F})$ and $U(n)$ have respective subgroups $H = \operatorname{SL}(n,\mathbb{F}),\operatorname{SO}(n,\mathbb{F}),\operatorname{SU}(n)$ and $K = \mathbb{F}^\times,\mathbb{Z}/2,U(1)$.
The fundamental group of the Klein bottle is $G = \langle x,y \mid xyx = y \rangle$. This is just the nontrivial semidirect product of $\mathbb{Z}$ with itself. Interestingly, the other (trivial) semidirect product $\mathbb{Z}^2$ is the fundamental group of the other closed surface of Euler characteristic $0$, namely the torus.
The affine group $A(n,\mathbb{F}) = \mathbb{F}^n \rtimes \operatorname{GL}(n,\mathbb{F})$. Its elements are transformations $\mathbb{F}^n \to \mathbb{F}^n$ of the form $x \mapsto Ax + b$, with $A$ an invertible matrix and $b$ a translation vector. The exact sequence is $$1 \to \mathbb{F}^n \to A(n,\mathbb{F}) \xrightarrow{f} \operatorname{GL}(n,\mathbb{F}) \to 1$$ where $f$ forgets the affine structure (the translation part).
The hyperoctahedral group $O(n,\mathbb{Z})$ is the group of signed permutation matrices. We have two decompositions $O(n,\mathbb{Z}) \cong \operatorname{SO}(n,\mathbb{Z}) \rtimes \mathbb{Z}/2$ and $O(n,\mathbb{Z}) \cong (\mathbb{Z}/2)^n \rtimes S_n$. In the corresponding exact sequences the surjective map is respectively the determinant homomorphism and the "forget all the signs" homomorphism.