Which court is called a court of record?

The Supreme Court and High Courts are called a court of record. 

Supreme Court as a Court of Record

As a Court of Record, the Supreme Court has two powers:

• The judgements, proceedings and acts of the Supreme Court are recorded for perpetual memory and testimony. These records are admitted to be of evidentiary value and cannot be questioned when produced before any court. They are recognised as legal precedents and legal references. 
• It has the power to punish for contempt of court, either with simple imprisonment for a term up to six months or with a fine or with both. 

High Court as a Court of Record

As a Court of Record, the High Court has two powers:

• The judgements, proceedings and acts of the High Courts are recorded for perpetual memory and testimony. These records are admitted to be of evidentiary value and cannot be questioned when produced before any subordinate court. They are recognised as legal precedents and legal references. 
• It has the power to punish for contempt of court, either with simple imprisonment or with a fine, or with both. 

Further Reading:

• High Courts of India
• Supreme Court of India