An emergency fund is self-funded insurance for an unplanned loss of income.
Also known as an eFund, an emergency fund usually consists of cash reserves valued at 3-6 months of basic living expenses. Like insurance, you don't touch it until you need it, such as after a loss of job or a new/sudden disability that renders you unable to work for substantial-enough pay to meet your current needs. Note that "basic living expenses" are usually considered to consist of the minimum needed to pay for the essential monthly bills -- including mortgage or rent, auto payments, consumer-debt minimum payments, utilities and food.
Since an Emergency Fund is pretty much intended to sit idle until you need it, many Fools stash it in a high yield money market account at a banking institution of their choice. (If you are interested in opening such an account, you may be able to get a referral from another Fool through which you both may get a bonus interest payment.)
Many folks do not have enough cash to immediately fund an Emergency Fund, so they establish a savings goal and make regular monthly payments. In addition to establishing a savings goal for an Emergency Fund, some Fools establish other savings goals, such as household funds, vacation funds, auto funds, home downpayment funds, engagement funds, etc.
While some may treat an emergency fund as more of a household fund or petty cash fund -- dipping into it for short-term unexpected expenses -- a classic emergency fund is never touched until that "rainy day" comes, complete with drenching floods. The danger with using an emergency fund for short-term expense "loans" is that you may not be as diligent at paying yourself back once the expense has been covered. Then, when the true rainy day arrives, you are left with nothing but a torn and broken umbrella with which to battle the onslaught.
One size does not fit all in the emergency fund area. You need to adapt it to your particular situation. What kinds of events are most likely to come up, what costs are likely, and how will you deal with them?
Examples: Your car breaks down. A family member is ill and you must make a quick trip to see them. Your television is stolen. Your apartment is flooded. You break your leg and can't work for months. etc etc
If you have equity assets, usually you can arrange to cover things given a bit of time, but you need something to cover that first month or so. A credit card can be enough in some circumstances. Some use bond funds. Many have check writing that let you simply write a check when you need cash. The funds earn interest until the check clears. Carrying a balance in an interest bearing checking account or in a passbook saving account or money market account used to be recommended, but with interest rates so low, those choices are almost painful.