A court of equity compels actions rather than awarding damages - although they can be permitted to award damages. For example, the court might decide which house a child lives in, and what is a fair distribution of marital assets - not based on "damage". Or issue restraining orders. But this is done according to law. Equity doesn't mean "this court doesn't follow law", it is related to the kind of relief they primarily award. A court of law deals with things that can be quantified in terms of monetary damages (primarily). This is not a criminal/civil law distinction - it primarily is a distinction in civil court, but there is some overlap - such as when violent acts are considered in a family court.
This all comes from England centuries ago. English courts of law awarded monetary damages (including things like land) and could be appealed to the king. They developed courts of equity that the church ran to decide issues of fairness.
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It means that in certain situations , principally custody and visitation that there are overriding principles of fairness that triumph strict rules of law. In matters involving children there are many technical rules of law while at the same time Judges will always try to make Orders that are in the best interest of a child. That same concept creeps into spousal support litigation and occasionally in regards to reimbursement claims involving property issues.
The prior answers are correct; to make address what it means for you, is that the family court is charged with deciding matters that canno be judged in terms of contract or money. The family court judges are deciding issues relating to your divorce, how both parties financially survive after the divorce, what happens to the children, how do the parents share time with each other, who pays for what, who gets the property. There are also situations where people live together but are not married, what happens to them and their children? Equity should mean fairness, sometimes the decisions do not seem fair to one person or the other. Although familiy court is supposed to be fair and equitable, those are the principals. Often, the rules are complicated, and the laws are not known by the people whose lives are being decided. If you are thinking of navigating family court to judge your dispute, you should at the least consult with a lawyer first. Equity and fairness should come to the decisions that you receive from the court, but they are often hard to effectuate. Good luck to you.
Disclaimer: The information you obtain at this answer is not, nor is it intended to be, legal advice. You should consult an attorney for individual advice regarding your own situation. Review of this answer does not in any way constitute legal representation,