Conductor is the term used for wires that are in a cable. For example a 3 conductor romex cable has 3 separate wires inside the cable. The wires insulation are typically color coded for the sole purpose of identifying one wire from another. Now in residential wiring, it's very common that one of those wires is just a bare copper wire with paper sheathing around it for protection as well as isolation from the other wires. This wire is used as a ground wire and is typically not counted as a conductor since most residential wiring calls for a ground wire at all times.
The conductors inside the cable come in a variety of sizes. The purpose of this is due to the amount of current which has to travel through the conductor in order to operate the load. For example, a circuit operating a single light bulb will only carry a small amount of current because the light bulb only draws 1/2 an amp. But if that same circuit were to operate 100 light bulbs, now that circuit needs to carry 50 amps current. This would require a larger conductor size to handle the heat that is created. Therefore proper wire sizing is critical.
Most 120Vac household circuits are fine using a 12 or 14 gauge wire. The smaller the gauge Number the larger the conductor size.
So when your needing a three conductor cable to wire up a basic 2-way switch, you will need a 14/2 w/ground cable. What that number means is 14 gauge (wire size), 2 conductor which also includes a bare ground wire. The 2 conductor only refers to the insulated wires and not the ground.
So 14/3 w/ground represents 14 gauge wire, 3 insulated conductors and 1 bare copper ground wire.
Now some wire manufacturer's will put a green insulation on their ground wires but then it will typically count as a conductor. Kinda confusing isn't it. But try not to put too much thought into it. Most common house circuits are very basic and thats all we really care about for now.