The electron temperature and density are measured from spectral lines of atoms and ions using methods of so-called emission spectroscopy.

The temperature is determined from a Boltzmann plot where measured line intensities are used. It is a long story to describe it in detail in a short post. Even wiki article is rather vague.

The temperature of heavy particles is another story, but when plasma is sufficiently dense it may be close to electron temperature.

@Jacek, thanks for the reply. Do you have a journal article where I can read how spectral lines are measured and how line intensities are used to determine the electron temperature?

The electron density of laser plasma can be measured using holographic interpherometry. General insructions can be found in "Application of holographic interferometry for plasma diagnostics" by A N Zaidel', SOV PHYS USPEKHI, 1986, 29 (5), 447–466

https://www.turpion.org/php/paper.phtml?journal_id=pu&paper_id=3381

or in

Article Holographic interferometry of physical processes - The scien...

There are two review articles I would like to recommend to you.

"Zhang S, Wang X, He M, et al. Laser-induced plasma temperature. Spectrochimica Acta Part B: Atomic Spectroscopy, 2014, 97: 13-33." and "Harilal S S, Phillips M C, Froula D H, et al. Optical diagnostics of laser-produced plasmas. arXiv preprint arXiv:2201.08783, 2022."

The reply by Jacek Hoffman is satisfactory as a short post. Still, I would like to drow the attention to the fact that the plasma is very often outside LTE and therefore an adapted Collisional-Radiative (C-R) model or Detailled Global Model (DGM) have to be used in order to compare theoretical to experimental line intensities in OES.