A dissenting opinion is one issued by a judge who disagrees with what the other judges on the panel decided. At both the appellate court level and the supreme court level in the United States, there are multiple judges who decide a single issue. The decision of the majority is the rule of law that applies in the case, and under the doctrine of stare decisis, the decision of the majority becomes the law that applies in all similar situations unless or until a court at the same level or a higher level overturns or overrules the opinion.
When the panel of judges makes a decision about how a given law should be interpreted or applied, they do not just write the specific answer of how it should be applied. Instead, they write an opinion explaining their reasoning and the way in which they came to a decision. This is done so other judges, as well as law students, legal scholars, experts and individuals, can use the information to get an indication of how the law will apply to them. For example, if a judge was deciding a case about whether a scooter constituted a vehicle for purposes of a statute prohibiting vehicles in the park, the court would explain its reasoning about exactly why they decided a scooter did constitute a vehicle in that particular case. Then, someone who wondered whether a bike also constituted a vehicle could look at that reasoning to decide whether or not it was likely the court would also classify a bike as a vehicle.
The majority opinion is published in case books and becomes law, but the dissenting opinions of the judges who don't agree with the majority are also published. In the dissenting opinion, the judge or judges who disagreed with the majority will write about exactly why they disagreed with the majority. They may also include information about how they would have interpreted the law and decided the case.
The dissenting opinion is important in case a later court wants to come along and overturn or overrule the verdict. A verdict or decision can be overturned by a court at the same level or higher. For example, another later appellate court could decide that the judges had gone wrong somewhere and overturn the method by which a given law is interpreted. A verdict can only be overruled by a higher court though; for example, the Supreme Court of the United States can overturn a decision made by the appellate court. When these later courts read the case and decide what to do, they can consider the dissenting opinion to get a more fuller view of the manner in which all of the judges in the court looked at the law, and can then decide what makes the most sense.