This question and this other one explain the differences between 会{かい}話{わ} and other words related to "conversation," but they don't cover 対{たい}談{だん}, which I've also learned has the same meaning.
Seen as both 会話 and 対談 are defined as conversation, dialogue, and talk, what is the difference between them? When should I use one over the other?
対談 means a discussion between two people which is usually set up as a special occasion and whose topic is set in advance. You can think of something like panel discussion (without panels). 議論 is a general term for discussion, and can be used for discussion in ordinary meetings, for example and there is no implication on the number of people. In this sense, 対談=議論+eventness+two people.
A discussion event involving three people is called 鼎談{ていだん} (after a three-leged pot 鼎{かなえ}). A standard word for panel discussion is 座談{ざだん}(会{かい})
会話 is a general term referring to any type of conversation that happens in daily life.
会談{かいだん} is a word similar to 対談. I'm not so sure how they are distinguished, but my impression is that 会談 is more of a talk between politicians with less specific topics.
対談, as least in modern Japanese, almost specifically refers to a talk session between two celebrities in the format of a talk show or a (mutual) interview. The topic is generally casual and not very socially important (e.g., their personal work style or motto), and hot debating is not intended. In a typical 対談, the two celebrities talk with each other, sometimes like an interviewer and sometimes like an interviewee. 対談 is a very common format of a magazine article.
Here is a typical 対談, one between a Nobel Prize winner and a shogi (Japanese chess) champion: ノーベル賞学者と史上最年少四冠のスペシャル対談!!
The two people selected for a 対談 are typically unrelated in their everyday work. Sometimes, there is an unnamed third person who plays the role of a host and provides various topics, but the main actors are always the two celebrities. If one of the two is an unknown writer, it is simply called an インタビュー. If three or more celebrities are present, it is usually called a 座談会. When politicians and such engage in serious discussion, it is called a 会談.
会話 is just "conversation"; this is something ordinary people do every day.
