As synonyms, these words are different mainly in their connotations: Positive, neutral, or negative.

Normal: Has neutral or good connotations. The opposite is weird, strange, abnormal. “Mary is always meeting weirdos. She would love to meet a normal guy for once.”

Regular: Has neutral connotations. The opposite could be special, rare, or unusual. In reference to a person, “regular” is usually a compliment, meaning unpretentious and down-to-earth. “Joe is a regular guy. You’ll like him.”

Ordinary: Has neutral or negative connotations. The opposite would be extraordinary, special, unique. In reference to a person, “ordinary” can easily be an insult. In American English, the word “ornery” was originally a contracted form of “ordinary”, and has its own meaning (which is rather negative):