Emergency Savings
An emergency savings account is money that is easily accessible and is used to cover living expenses if you lose your job or sustain injury that keeps you out of the office. Business Insider recommends having anywhere from 3-8 months of salary stashed away [1], but it can never hurt to have more.
The emergency fund will be used to cover all of your expenses until you return to work (housing, auto, insurance, food and drink) so the actual amount of money you need to save will be unique to your situation.
Car or House Repairs
While some expenses can occur completely unforeseen, others can be planned for even if they aren't imminent. For example, if you drive a car that is no longer under warranty, you may want to have an extra $1,000 or so saved in case anything happens. This amount will cover a variety of expenses, such as new tires or brakes, and at the very least is a start if you find yourself faced with a more expensive repair.
If you own a house, you'll want to handle any repair work as soon as possible. Doing so will help maintain the value of house, which is an investment after all. You won't want to wait 5-6 months if you discover that you have a plumbing issue; the result of waiting could be catastrophic.
Life Insurance
This only comes into play if the absolutely unthinkable happens, but life insurance is something you may not think about having until it is too late. Setting aside just an extra $50 [2] should ascertain you a policy that will make sure that your spouse and kids have a house, food, and money for their other needs if you aren't there to provide for them.
There are so many other unseen money pits that can appear out of nowhere, and it is impossible to be able to plan for everything. Having money set aside will give you peace of mind about life's daily obstacles and will keep you from sliding into debt.
Sources:
[1] Here's how to figure out how much emergency savings you need, Business Insider
[2] How much will life insurance cost me, CNN Money
NMLS #407985