It's true that at high concentrations, like the nearly 100-percent pure alcohol used in sterilizing solutions, alcohol can indeed kill cells and neurons (and nearly anything else). But given that the blood reaching your brain is only at 0.08 percent alcohol if you're legally intoxicated, you're not actually killing any cells.
Don't believe it? A major study by Grethe Jensen and colleagues in 1993 matched brain samples taken from both alcoholics and nonalcoholics, from groups of the two dead from non-alcohol-related causes. There were no significant differences found in either the number or density of brain cells between the groups.
What alcohol can and does do to your brain is affect the way your neurons get their firing triggers from glutamate. It infiltrates the glutamate receptors in your synapses, hurting their ability to send off their normal "fire" messages. Alcohol has this impact all across your brain—the parts that control muscles, speech, coordination, judgement, and so on. Keep that in mind the next time you or someone else claims that they drive, golf, or otherwise perform some task better with alcohol's help.
Feel free to read the entire article about what alcohol does and doesn't do over at LifeHacker .