= Solution
The electron <Debye length> is the scale over which a weak electrostatic disturbance is screened. Take the ions as an immobile neutralizing background and the electrons as an isothermal <Boltzmann distribution>. For a small potential $\phi$,
$$
n_e(\phi)=n_0e^{e\phi/(k_BT)}\simeq n_0\left(1+\frac{e\phi}{k_BT}\right),\qquad \rho\simeq-\frac{n_0e^2}{k_BT}\phi.
$$
<Poisson's equation> then becomes $\nabla^2\phi=\phi/\lambda_D^2$ away from the perturbing source, with
$$
\boxed{\lambda_D=\sqrt{\frac{\epsilon_0k_BT}{n_ee^2}}.}
$$
A test charge has a screened potential proportional to $e^{-r/\lambda_D}/r$. For fully static screening with mobile ions, inverse squared Debye lengths from all species add; the displayed expression is the electron scale used below.
The electron <plasma frequency> is the natural collective oscillation frequency when electrons are displaced from a nearly fixed ion background. Displace a uniform electron fluid by $\xi$ along a slab normal. The charge separation produces the restoring electric force $-n_ee^2\xi/\epsilon_0$ on an electron. Hence $m_e\ddot\xi=-n_ee^2\xi/\epsilon_0$ and
$$
\boxed{\omega_p=\sqrt{\frac{n_ee^2}{\epsilon_0m_e}}.}
$$
It sets a response time $1/\omega_p$ and the cold unmagnetized electromagnetic cutoff frequency. These collective effects arise from long-range electric fields, even when individual binary collisions are infrequent.
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