Ligand-gated refers to the fact that the ion channel opens when a ligand binds to the receptor on the channel. The ligand in this case would be the neurotransmitters sent from one neuron to another. So one neuron has an action potential going down it, it reaches the end of the nerve and the nerve then sends neurotransmitters across the synapse. These action potentials then bind to the start of the next neuron, activating these ligand-gated channels and allowing a small influx of sodium which brings the membrane potential above the threshold potential, allowing the voltage-gated channels to open and allowing a massive influx of sodium. This sodium then travels down the neuronal cell body and changes the membrane potential in the next section of nerve to be above threshold potential, thus perpetuating the cycle all along the nerve until it reaches the synapse.
So basically, the ligand-gated channels are only implicated in the very first section of nerve. From then on, that initial influx of sodium travels down the cell body to stimulate voltage-gated channels to perpetuate the rest of the neuron.
Regarding the 'all or nothing' effect, it refers to the fact that unless membrane potential reaches above the threshold, then nothing will happen. In addition, regardless of how much you reach above resting potential, there's no difference in the strength of the action potential. Basically, if you go above threshold, all of the voltage-gated channels will open to achieve a set membrane potential. If it doesn't go above threshold, no action potential is generated.