While glutamate is a non-essential amino acid, it is also a very potent neurotransmitter. When excess glutamate is present it over-stimulates NMDA and other receptors, which trigger intracellular signaling cascades initiating apoptosis and other death pathways. One school of thought is that this happens as a safety switch to prevent excessive nerve firing which could lead to even worse consequences other than just losing the stimulated neuron.
There are many reviews on this topic, one of which is PMID: 11733308. I think it nicely will describe what you are looking for.
Not every cell is sensitive to glutamate, as Adam pointed out certain receptors such as NMDA-receptors are required to make a cell susceptible to glutamate. The current dogma is that excessive stimulation of NMDA-receptors causes calcium overload which then triggers cell death.
Is there any universal and specific blocker for neuronal excitotoxicity? Dr. Maximilian Peters has mentioned that NMDAR is needed for neuronal excitotoxicity. Is it really so? Many literature has suggested the role of other ca conducting receptors? Amoung the NMDAR subunit combinations, It is well reported that NR1/NR2A has a protective role and NR1/NR2B subunit is causing excitotoxicity.
Whether excitotoxicity will happens due to ions other than calcium?
At very high concentrations (up to 1 mM) glutamate may also inhibit the antiporter Xc- which exchanges cystine against glutamate. This inhibition leads to a reduction of the intracellular content on glutathione and hence to an oxidative shock leading to neuron apoptosis (see the paper published by P Albrecht, J Lewerenz, S Dittmer, R Noack, P Maher, A Methner: mechanisms of oxidative glutamate toxicity: the glutamate/cystine exchange antiporter Xc- as a neuroprotective drug target CNS Neurol Disord drug Targets 9, 2010, 373-382. see also P Froissard, H Monrocq, D Duval. Role of glutathione metabolism in the glutamate-induced programmed cell death of neuronal-like PC 12 cells Eur J Pharmacol 326, 1997, 93-99).