The difference between an Emotional Support Animal (ESA) and a "normal" pet is not so much about the animal as the person. Essentially an ESA is a pet where their owner gets a benefit towards a specific emotional or psychological condition as a result of having them as a pet.
The animal has no specific training unlike a service animal which will be trained to do a specific task for their owner.
When it comes to "rights" regarding ESAs - well that will vary by jurisdiction. In the US for example it can mean that the owner can get exemptions from housing and travel restrictions on the pet where the animal is classed as an ESA - in order for a pet to be classed as an ESA the person needs to suffer with a diagnosed emotional or mental condition that a health practitioner will certify the animal provides a significant benefit for.
In the UK ESAs don't enjoy similar legal protection/status - there are some private registries that some businesses will agree to honor but this is purely voluntary, nothing gives ESAs (or their owners) any additional rights over a normal owner and pet.