When does a hypersurface have contact-type?

Not all hypersurfaces in $\mathbb{R}^{2n}$ are of contact type.

Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=\frac{1}{2}\vert p\vert^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.

Here is an example of a surface which cannot be made of contact type even after isotopy. The elliptic surface $E(1)$ is obtained by blowing up $\mathbb{C}\mathbb{P}^1$ nine times. It is a symplectic four-manifold with fibre of genus 1 and 12 singular fibres. If we make a fibred connected sum between two copies of E(1) we obtain the elliptic surface $E(2)$, which is a symplectic four-manifold with fibre of genus 1 and 24 singular fibres.

Then $E(2)$ is separated by a torus $T^3$ into two pieces which are diffeomorphic to the complement of a regular fibre in $E(1)$. This shows that the separating torus cannot be made of contact type because, by a result of Chris Wendl, all strong fillings of $T^3$ are diffeomorphic to a blow up of $T^*T^2$.

There are also examples that are characterized dynamically. By Viterbo, every compact, contact-type hypersurface of $\mathbb{R}^{2n}$ carries a closed Reeb orbit, i.e. a closed leaf of the characteristic line bundle. (Actually, Hofer and Zehnder show this is true for any stable hypersurface.) There are examples of autonomous Hamiltonians on $\mathbb{R}^{2n}$ that fail to have any closed orbits on a specific level set, thus providing examples of hypersurfaces that fail to be stable. These are due to Ginzburg and Gurel.