Maybe it's useful to add a few further clarifications to the original question, which Will has answered for solvable or nilpotent algebraic groups.
The basic definitions are the same over any algebraically closed field $K$, though the label "reductive" may be misleading since the group need not act completely reducibly in a finite dimensional representation. But in the three textbooks with the same title (by Borel, Springer, and myself), the theory is developed under the restrictive assumption that $G$ is a connected linear algebraic group. This was the focus of the Chevalley classification and already involves considerable work. When $G$ is allowed to be disconnected (including finite matrix groups), the same basic outline can be followed. But then the extension of the identity component relative to the finite quotient $G/G^\circ$ need not split and is often delicate to sort out.
In the definition of almost direct product for an abstract group, Borel (in his preliminaries) specifies a group with two or more normal subgroups, such that the product map is a surjective group homomorphism with finite kernel. In particular, as xuhan comments, the subgroups commute with each other..
It's helpful to have in mind some natural examples of disconnected reductive groups. For instance, take $G$ to be connected and semisimple, with a maximal torus $T$ and its normalizer $N$ in $G$. Then $N$ fails to be connected, but has $T$ as identity compoennt and quotient equal to the Weyl group. This example already indicates the tricky nature of almost direct products, explored by Tits and others: see my references in my earlier answer here. Note that $N$ doesn't need to be solvable but does admit a suitable almost direct product decomposition. Another class of examples occurs when $G$ fails to be simply connected; then the centralizer in $G$ of a semisimple element is always reductive but need not be connected. However, the Borel-Tits description of such centralizers shows how to describe the part outside the identity component in terms of a subgroup of the Weyl group.