Yes, an instanton is a classical solution to the Euclidean equations of motion with finite action. Its topological charge is given by $k = \frac{1}{8\pi}\int \mathrm{tr}(F\wedge F)$ which is the integral of the divergence of the Chern-Simons current.
There are many different instantons possible. A generic instanton for $\mathrm{SU}(2)$ and topological charge 1 is given by the BPST instanton
$$ A_\mu^a(x) = \frac{2}{g}\frac{\eta_{\mu\nu}^a(x-x_0)^\nu}{(x -x_0)^2 - \rho ^2}$$
where $x_0$ is the "center" of the instanton and $\rho$ its scale, also called the radius. The $\eta$ is the 't Hooft symbol.
A large class of instantons of topological charge $k$ may be described as follows: Transforming the BPST instanton by the singular transformation $x^\mu \mapsto \frac{x^\mu}{x^2}$ leads to the expression
$$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\frac{\rho^2}{(x-x_0)^2}\right)\right)$$
for the transformed instanton, and one now makes the more general ansatz
$$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\sum_{l=1}^k \frac{\rho_l^2}{(x-x_{0,l})^2}\right)\right)$$
which leads to an instanton solution of topological charge $k$. This construction can be generalized to other non-Abelian gauge groups.
The generic construction of all instantons on four-dimensional spacetimes of gauge group $\mathrm{SU}(N)$ is given by the ADHM instanton, see also the original paper "Construction of instantons" by Atiyah, Drinfeld, Hitchin and Manin.
