If $G$ is an algebraic group (say over $\mathbb C$), acting on an algebraic variety $X$, the orbits are always locally closed smooth algebraic subvarieties of $X$. This is standard, and easy to prove. If $x$ is a point of $X$, and $G \to X$ is the morphism sending $g$ to $gx$, consider the orbit $Gx$ as a subset of its Zariski closure $\overline{Gx}$. By Chevalley's theorem, $Gx$ contains a dense open subset of $\overline{Gx}$. Hence, by homogeneity it is open andin $\overline{Gx}$, and smooth.
From this it is easy to deduce that all the orbits of $\mathrm{GL}_n(\mathbb R)$ in $\mathrm{M}_n(\mathbb R)$ are embedded submanifolds. The point is that if two real matrices are conjugate as complex matrices, then they are conjugate as real matrices. These means that the orbits of the action of $\mathrm{GL}_n(\mathbb R)$ in $\mathrm{M}_n(\mathbb R)$ are of the form $\Omega(\mathbb R)$, where $\Omega$ is an orbit of $\mathrm{GL}_n(\mathbb C)$ in $\mathrm{M}_n(\mathbb C)$; since $\Omega$ is a smooth algebraic variety, it follows from the implicit function theorem that $\Omega(\mathbb R)$ is an embedded submanifold of $\mathrm{M}_n(\mathbb R)$.