A linguistic remark: "Instantons" are the same mathematically to "solitons", particle-like solutions of classical field theories (explaining the suffix "on"). Unlike solitons, instantons are structures in time (explaining the prefix "instant").

A mathematical remark (using Donaldson's book on Yang-Mills Floer homology, appendix C of section 2.8), which supplements Igor Khavkine's answer:

Consider the Yang-Mills equations over the (3+1)-dimensional spacetime $Y\times\mathbb{R}$, using the Lorentzian metric $dy^2-dt^2$ (this is the "real life" physical picture). Yang-Mills solutions are solutions to the Euler-Lagrange equations $d_A^\ast F_A=0$ of the Yang-Mills functional $\int_{Y\times\mathbb{R}}(|E|^2-|B|^2)$, where we have decomposed $F_A=\ast B+E\wedge dt$. These solutions can be viewed as paths $[A_t]$ in the configuration space $\mathcal{B}_P$ of (gauge equivalence classes of) connections on the principal bundle $P\to Y$. In this viewpoint, $B$ is the curvature of $A_t$ (on $Y$), and $E$ is the velocity vector of the path $[A_t]\subset\mathcal{B}_P$, and the Yang-Mills functional is thus $\int(||\nabla_tA_t||^2-V(A_t))dt$ with $V(A_t)=\int_Y|F_{A_t}|^2$. *That means the 4-dimensional Lorentzian Yang-Mills solutions can be regarded as the motions of a particle moving on $\mathcal{B}_P$ in the potential $\int_Y|F_A|^2$.*

However, instantons are Yang-Mills solutions for the Euclidean metric. In the above picture, that means we need to reverse the sign of the potential, and we lose our physical description of particles. By the way, so far we have been describing the first paragraph of Igor Khavkine's answer. Let's move on to his second paragraph:

If we are to relate instantons with a physical description of particles, then we need pass to quantum mechanics on $\mathcal{B}_P$. We look for wavefunctions that are energy eigenstates for the potential $V=\int_Y|F_A|^2$, i.e. solutions to Schrodinger's equation on $\mathcal{B}_P$. Instantons will approximate these solutions. If the energies are greater than $V$, we have our usual classical picture of a ball rolling over a hill, but if the energies are less than $V$ ($E_0<V$) then we have "quantum tunnelling". Clarifying, the "leading order" approximation of Schrodinger's equation (the instantons) in these classically inaccessible regions will be given by *trajectories of particle motions on $\mathcal{B}_P$ with energy $-E_0$ in the potential $-\int_Y|F_A|^2$.*

This is really cool... In the toy model of a double-well potential (see *Wikipedia's article on instantons*), instantons are the solutions which tunnel from well to well. Mathematically that means we have a path $[A_t]\subset\mathcal{B}_P$ of connections which are asymptotic to flat connections on both ends (as $t\to\pm\infty$), and this is the setup for Yang-Mills Floer homology!