Court of first instance
The term 'juzgado de primera instancia' refers to a specific level of court in some Spanish-speaking regions. It is typically where a legal case will start and where decisions will be made regarding civil and criminal matters. The judge at this level usually takes statements, looks over evidence, and decides whether a case needs to go to a higher court or not. In English, this term translates to 'court of first instance'.
The court of first instance determined that the defendant was innocent.
This sentence is saying that the first court to hear the case has determined, or made an official decision, that the person who was accused or charged with a crime was in fact not guilty.
My father is the lawyer of the court of the first instance.
In this sentence, the person is indicating their father's profession. Their father is a lawyer at a court of first instance, which is the first and usually the lowest in grade of courts. Lawyers in this court are responsible for representing parties in trials.
I'm going to appeal the decision of the court of first instance.
The phrase is used here to indicate that the speaker intends to challenge, or appeal, the decision made by the court of first instance, presumably because they disagree with it and think it should be overturned.