What must a neurotransmitter cross in order to excite or inhibit an action potential in a receiving neuron?

Most neurotransmitter molecules get released from axon terminals, and the cross the synaptic cleft through diffusion. They then bind to receptors embedded in the membrane of the postsynaptic cell.

Neurotransmitters do not directly excite or inhibit action potentials per se. They can raise or lower the potential of the neuron (or more accurately, the region of the neuron near the receptor binding site). If the potential of the neuron (or a region of a dendrite containing active channels) exceeds the effective spiking threshold, then an action potential is generated.

Crash course has some good videos illustrating all this.

Reference: Yohan John