The electrical conductivity $\sigma$ relates the current density $\vec{j}$ to the electric field $\vec{E}$,

$$\vec{j} = \sigma \vec{E},$$

where the electrical conductivity depends on the electron density $n$ and the electron mobility $\mu_e$,

$$\sigma = ne\mu_e = \frac{ne^2\tau}{m^*}.$$

For metals, $n$ remains constant as the temperature changes. The mobility can be expressed in terms of the scattering time $\tau$ and the effective mass $m^*$. The electrons scatter from phonons, other electrons, and defects in the crystal with the scattering times $\tau_{\text{ph}}$, $\tau_{\text{e-e}}$, and $\tau_{\text{defect}}$, respectively. The total scattering time is shorter than any of the individual scattering times,

$$\frac{1}{\tau} = \frac{1}{\tau_{\text{ph}}} + \frac{1}{\tau_{\text{defect}}} +\frac{1}{\tau_{\text{e-e}}}.$$

This is known as Matthiessen's rule. The scattering time for defects is temperature independent, but the scattering times for electron-phonon scattering and electron-electron scattering get shorter as the temperature increases. Matthiessen's rule can also be written in terms of the resistivity $\rho = 1/\sigma$,

$$\rho = \rho_{\text{ph}}(T)+\rho_{\text{e-e}}(T)+\rho_{\text{defect}}.$$

For metals that can be reasonably well described by the free electron model (Ag, Al, Au, Cu, K, Na, Pt), the resistivity is dominated by defect scattering at low temperatures and electron-phonon scattering at higher temperatures. For these materials, the phonon component is proportional to temperature $\rho_{\text{ph}}\propto T$. Near room temperature the resistivity is often modeled as $\rho = \rho_{\text{ref}}(1 + \alpha (T - T_{\text{ref}})$ where $ \rho_{\text{ref}}$ is the resistivity at some reference temperature $T_{\text{ref}}$ and $\alpha$ is the temperature coefficient. A table of resistivities can be found on Wikipedia.

There are materials called heavy fermions where the effective mass is 100 - 1000 times the free electron mass and the electron-electron scattering dominates. Some examples are CeIn₃, CeCoIn₅, URu₂Si₂, UPd₂Al₃, and YbBiPt.

Semiconductors
Both electrons and holes contribute to the electrical conductivity of semiconductors,

$$\sigma = ne\mu_e + pe\mu_h.$$

Here $\mu_e$ is the mobility of the electrons and $\mu_h$ is the mobility of the holes. For an intrinsic semiconductor, the mobilities decrease with increasing temperature due to more scattering and shorter scattering times. However $n$ and $p$ increase exponentially as the temperature increases,

$$n = p = \sqrt{\frac{\pi D_cD_v}{4}}(k_BT)^{3/2}\exp(−E_g/(2k_BT)).$$

Consequently, the electrical conductivity of an intrinsic semiconductor increases as the temperature increases.