One of the perhaps most important theory of anaesthesia is that:
Most anaesthetics enhance the activity of inhibitory GABAa receptors
and other cys-loop ligand-gated ion channels. Other important effects
are the activation of a subfamily of potassium channels (the two-pore
domain K+ channels) and inhibition of excitatory NMDA receptors
Almost all anaesthetics (with the exceptions of cyclopropane, ketamine and xenon) potentiate the action of GABA at GABAa receptors (1)
At the cellular level, the effects of anaesthetics are to enhance
tonic inhibition (through enhancing the actions of GABA), reduce
excitation (opening K+ channels) and to inhibit excitatory synaptic
transmission (by depressing transmitter release and inhibiting
ligand-gated ion channels).(2)
- There is some controversy about whether or not xenon potentiates GABAa responses but at present the weight of evidence suggests it does not.
- Ketamine is believed to act by blocking activation of the NMDA receptor. (NMDA receptors are also an important site of action for anaesthetics such as nitrous oxide, xenon )
$\ce{GABA}$ receptors are ligand-gated Cl− channels made up of five subunits (generally comprising two α, two β and one γ or δ subunit).
Anaesthetics can bind to hydrophobic pockets within different GABAa receptor subunits.
Specific mutations of the amino acid sequence of the α subunit inhibit the actions of volatile anaesthetics but not those of intravenous anaesthetics, whereas mutations of the β subunit inhibit both volatile and intravenous anaesthetics.
This suggest that volatile anaesthetics may bind at the interface between α and β subunits (analogous to benzodiazepines that bind at the interface between α and γ/δ subunits:

Putative anaesthetic binding sites on GABAa receptor subunits. [A] A model of the α1 subunit of the GABAa receptor with a molecule of isoflurane shown sitting in a putative binding site. The transmembrane α-helices (TM) are numbered 1–4. [B] A model of the β2 subunit of the GABAa receptor with a molecule of propofol shown sitting in the putative binding site. (Adapted from Hemmings HC et al. 2005 Trends Pharmacol Sci 26, 503–510.)
References:
Olsen, R.W., Li, G.D., 2011.GABAa receptors as molecular targets of general anesthetics: identification of binding sites provides clues to allosteric modulation. Can. J. Anaesth. 58, 206–215.
Schüttler, J., Schwilden, H., 2008. Modern anesthetics. Handb. Exp. Pharmacol. 182.
Rang and Dale’s Pharmacology 8th ed H P Rang et al
Goodman & Gilman's The Pharmacological Basis of Therapeutics, 12th Edition