An emergency fund is just as it sounds. A fund of money set aside just in case you really need access to money at short notice. Keep the money in a separate bank account with instant access.
As a general rule, the financial community feels that the amount you should have set aside should be equivalent to three months worth of your income. Some advisers would counsel you to keep six months of income aside. This might sound like a lot, but is really quite a sensible number.
For example, you might fall and injure yourself. If your company decides not to make sickness payments, or the expat life insurance plan you have has a deferred period (an automatic wait before they pay you) or you are self employed with no employer based sickness assistance, you will still need money to live on.
This means that when the kids need piano lessons you do not use the emergency fund. However, when the boiler breaks and the house is starting to freeze, you do use it.
The whole point of the money is this: life has a habit of doing nasty things and it is only a matter of time until the next shock arrives – when that shock comes you will be better prepared financially to deal with it. This will mean that you only have to deal with the nasty problem rather than the problem plus financial issues.