This is a rather basic, and open-ended question: in several branches of mathematics and physics, we make an effort to classify linear operators $A$, especially orthogonal or unitary operators, by whether or not they have unit-determinant $\det A = 1$, i.e. whether they are special or not.
E.g., the special orthogonal group $\operatorname{SO}(N)$ is a subset of the orthogonal group $\operatorname{O}(N)$ and the special unitary group $\operatorname{SU}(N)$ is a subset of the unitary group $\operatorname{U}(N)$.
At the most basic level:
- Why are we often more (or especially) interested in these special operators?
- When acting on a vector, which quantities do the special operators leave invariant?
- What, if any, additional constraints are made on e.g. the spectrum of such special operators?
This is a very broad topic, so I can only mention a few starting points, and I'll mostly just discuss the special linear groups, $$SL(n, \Bbb F) := \{A \in GL(n, \Bbb F) : \det A = 1\} .$$
Other special groups, e.g., $SO(n, \Bbb R)$ and $SU(n, \Bbb R)$, are intersections of a suitable parent group and the special linear group over the appropriate field, for example, $$SO(n, \Bbb R) = O(n, \Bbb R) \cap SL(n, \Bbb R) ,$$ and so inherit some features from both groups. More can often be said in particular cases.
For any field $\Bbb F$ the group $SL(1, \Bbb F)$ is trivial, by the way, and so its behavior is exceptional among special linear groups. For ease of exposition henceforth we'll take $n > 1$. I'm happy, by the way, to give suggestions for further reading about any of these topics if you're interested.
Geometry First, the group $SL(n, \Bbb R)$ acts transitively on $\Bbb R^n - \{ 0 \}$, so the only property of a vector in $\Bbb R^n$ preserved under the action of $SL(n, \Bbb R)$ is whether the vector is zero or not. (Throughout I mostly refer to the real special linear group, but most of what's said here applies just as well to special linear groups over some/all other base fields).
On the other hand, the action of $GL(n, \Bbb R)$ on $\Bbb R^n$ induces an action of the space of volume elements $\bigwedge^n \Bbb R^n$---this vector space is $1$-dimensional and so linearity implies that this action is given by $$A \cdot \omega = (\det A) \omega .$$ So, $SL(n, \Bbb R)$ consists precisely of the linear transformations that preserve a volume form, or, equivalently, both (unsigned) volume and orientation on $\Bbb R^n$. (Thus, for example, the special orthogonal group $SO(n, \Bbb R) = O(n, \Bbb R) \cap SL(n, \Bbb R)$ consists of linear transformations that preserve a inner product on $\Bbb R^n$ and a choice of orientation; these two objects determine a volume form which is thus also preserved.)
Just as the definition of the orthogonal group characterizes Euclidean geometry---namely, the geometry on $\Bbb R$ under transformations of preserving lengths (and therefore angles)---the special linear group characterizes the geometry on $\Bbb R$ under transformations preserving a volume form. This is called equi-affine geometry, and since $O(n, \Bbb R) \lneq SL(n, \Bbb R) \lneq GL(n, \Bbb R)$, an equi-affine structure contains strictly more information than an affine structure (in which only incidence is preserved---this corresponds to $GL(n, \Bbb R)$) but strictly less information than an orthogonal (Euclidean) structure.
Alternatively, since $SL(n + 1, \Bbb R)$ acts linearly and transitively on $\Bbb R^{n + 1} - \{ 0 \}$, it descends to an action on the real projective space $\Bbb R P^n := \Bbb P(\Bbb R^{n + 1} - \{ 0 \})$. Since $SL(n + 1, \Bbb R)$ maps $k$-planes to $k$-planes (in particular for $k = 2$, this action preserves projective lines, which are precisely the unparameterized geodesics of the canonical ("flat") projective structure on $\Bbb R P^n$. So viewed, projective space, equipped with the projective lines, serves as a model for projective (differential) geometry the same way that Euclidean space serves as a model for Riemannian geometry. Similar constructions give rise models of other, more exotic, but still interesting, geometric structures, realized as $SL(n + 1, \Bbb R)$-homogeneous spaces.
Algebra It turns out that $SL(n, \Bbb R)$ is exactly the commutator subgroup $\langle \{ A B A^{-1} B^{-1} : A, B \in GL(n, \Bbb R) \} \rangle$ of $GL(n, \Bbb R)$. This is closely related to the fact that the Lie algebra $\mathfrak{sl}(n, \Bbb R)$ of $SL(n, \Bbb R)$ is simple: Its only ideals are the Lie algebra itself and $\{0\}$. (In fact, $SL(n, \Bbb F)$ is very nearly simple: Its center is $Z := Z(SL(n, \Bbb F)) = \{\zeta I_n : \zeta \in \Bbb F, \zeta^n = 1\}$, and the quotient $PSL(n, \Bbb F) = SL(n, \Bbb F) / Z$ is simple, except for a few cases over small finite fields.) Simplicity has many powerful consequences (see the next section).
Since the determinant of $A$ is the product of the eigenvalues of $A$, by the way, the constraint on the spectrum $(\lambda_a)$ of $A \in SL(n, \Bbb F)$ is precisely that $\lambda_1 \cdots \lambda_n = 1$, but this places no constraint on any single eigenvalue $\lambda_a$ beyond $\lambda_a \neq 0$.
Representation theory The fact that $\mathfrak{sl}(n, \Bbb R)$ is simple makes available the well-developed theory of representations of (semi)simple Lie algebras and Lie groups, which applies just as well to the other groups you mention, $O(n, \Bbb R), SO(n, \Bbb R), SU(n)$, as well as others. In some ways, $SL(n, \Bbb C)$ is the easiest family of semisimple Lie groups to understand and so it makes for a natural first example when learning new concepts in the topic. This gives, among many other results, a complete description of all of the representations (in both the group and algebra settings), and complete reducibility of all representations, i.e., every $\mathfrak{sl}(n, \Bbb C)$-representation can be decomposed essentially uniquely as a direct sum of irreducible representations. (This theory can be bootstrapped to recover the representation theory of $GL(n, \Bbb C)$, which is similar to the representation of the special linear group, but there are some key differences: For example, since the special linear group preserves a volume form on $\Bbb C^n$, contraction with that volume form defines a natural isomorphism $\Bbb C^n \cong \bigwedge^{n - 1} (\Bbb C^n)^*$, but the general linear group preserves no such volume form, so $\Bbb C^n$ and $\bigwedge^{n - 1} (\Bbb C^n)^*$ are inequivalent as $GL(n, \Bbb C)$-representations.)