In general, you will be SOL.
Most tickets sold are non-refundable tickets which means there's little or no recourse if your travel plans change.
However, it depends on how you define emergency.
If it's something like a major weather disruption, earthquake, civil unrest, then the airline will typically work with you. In other words, if the airline cannot operate your flight due to said emergency, they'll work with you.
If it's your grandma sick in the hospital or you simply changed your mind, then no, the airline won't do anything.
If you buy a full fare ticket, you could get a refund. You can also buy insurance which MAY cover some emergency situations.
It all comes down to the type of ticket you buy.
Honestly, you would know if you had an eligible ticket as you would have paid a lot of money.
But if you by chance bought it on Southwest, you could at least get travel credit towards another flight