An emergency fund is a savings allotment that you set aside to cover costs in the instance of an emergency. An emergency can include a tornado hitting your house and you have to pay for unforeseen medical expenses (usually your deductible if you have insurance), your car needing a new transmission, needing to take unpaid medical leave for a few months due to surgery or unforeseen health issues. Any of these things warrants using your emergency fund.
When you are starting out, your emergency fund can be as small as $500 in savings, mine was only $1000 for the last 3 years, but this is money that you earmark for emergencies and not causal spending.
Causal spending goes in the budget and the first focus should be on setting aside money for your emergency fund. If you are struggling to pay the bills each money, then you absolutely need an emergency fund.
If you are struggling to find ways to save towards this fund, consider getting a second job, skipping eating out, cooking all your meals at home, and giving up a few luxuries so that you can save for your emergency fund. Remember, it is only temporary and you will not need to live like this forever, but make the sacrifice now so that you and your family can live a life of security and comfort moving forward.