According to this article, 40% of Americans CAN'T cover a $400 emergency expense which is really concerning. It isn't if but when an emergency will happen and the fix will require money. Just within the last year or so we have had several unexpected expenses, some of which would have been big problems if we hadn't had the savings to cover them including a dead hot water heater, a 30 day notice to paint all of the trim around the house by the HOA, plumbing issues, and kid issues (ie: a couple of kids had unexpected emergencies and didn't have the savings to cover the problem so we gave them the money).
Needless to say, having money in savings is critical for your peace of mind and well being. Every day people have financial issues which can range from minor (they don't have enough money for an impromptu lunch out with co workers) to major (they get sick, miss days at work, their paycheck is short, and they can't cover the rent which leads to eviction!).
The need for an emergency fund is so important, that even if you are swimming in debt, financial expert Dave Ramsey recommends that you put aside money in an emergency fund first to cover unexpected emergencies before doing anything else to get out of debt.
If you don't yet have an emergency fund, this should be your top priority. You can find several ways to quickly bulk up your emergency fund here, here, and here.