Yes, the relay would normally (but not always) switch just the live 120vac (hot) wire. It's a pretty simple device, coil voltage (12v in this case) switches the load connection. It could be normally off (NO - the load circuit is open unless coil voltage closes it), or normally closed (NC - the load circuit is closed unless the coil forces it open). As I understand it though, the load and use matters.

One factor is power needed to hold state. A mechanical relay takes power to hold "other" state (so a NC held open takes power to stop a load from running, a NO held closed takes power to run the load). As a general proposition, we don't care a lot about the current needed to keep a NO relay closed to run an opportunity load, but your use may be different.

Another is whether the load needs or can withstand rapid switching. You usually want to avoid bouncing between open and closed states (load comes on, voltage drops, controller drops control voltage, relay opens, battery voltage recovers, apply coil voltage, relay closes, load starts .... ). As I understand it, a solid state relay would be better if you wanted a "bouncy" relay (since the switch is done at the zero crossing of the ac wave), but in less " bouncy" applications there are advantages to a mechanical relay.

So it would help if we knew more about what you're trying to switch. Others here know far more than me about relays, but I think I know enough to know the load matters.