Your neurons communicate using neurotransmitters. Neurotransmitters are released from one neuron, and then bind to receptors on the next neuron.
Alcohol and binds to certain receptors (specifically GABA, acetylcholine, serotonin, and glutamate), which messes with which neurons get stimulated.
In the case of the GABA receptor, when alcohol binds, it makes the neuron less excitable (it does that by allowing more Cl- to flow into the cell, reducing the electrical activity).
Less excitable neurons --> less neural activity --> you're tipsy.
Have a great weekend!