What is the ruling on a wife who was divorced by intention, and her husband took her back that same night also by intention, without witnesses or uttering words, and what is the ruling on this divorce to begin with?

The majority of scholars hold that a conditional divorce takes effect upon the fulfillment of the condition, and this is the preponderant opinion. If you swore to divorce your wife if her father did not come to your house, and you set a specific time for this, and her father did not come before the expiration of the intended time, then the divorce is effective according to the majority.

However, Sheikh al-Islam Ibn Taymiyyah is of the view that an expiation for an oath (kaffarat yamin) is required if you did not intend a divorce.

According to the majority's view that the divorce takes effect, you may take her back before her waiting period ('iddah) is complete. The return (raj'ah) does not occur by mere intention according to the majority, but it does occur through sexual intercourse or its مقدمات (preliminaries) according to some scholars, even without intention. The return is valid without witnessing (ishhad) according to most scholars.

If you did not specify a time for her father to come, then the divorce does not take effect unless his coming becomes impossible, such as due to death. This is because the particle "إن" (if) is for condition and does not imply a specific time, except in that the action upon which it is conditional necessarily involves time.