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The three naming styles are the controls above the domain search box, and they change what the search does with what you type. Traditional and Brandable are a pair and you pick one of them. Creative is a separate switch and changes the job entirely.
Understanding them is worth a minute, because the commonest way to get confusing results is to leave a control set for one job and then ask the search for a different one.
Traditional, for names made of real words
This is the default, and it looks for names built from the words you typed. A plumbing company searching for its trade gets results that contain the word.
Choose it when what the business does is the thing people search for, or when a descriptive name is a genuine advantage. A name that says what it does needs no explaining, and it can help with the search terms it contains.
Its weakness is supply rather than quality. The good combinations in a popular extension were taken years ago, so a strict traditional search across many extensions often comes back with very little. If a word is crowded, modifiers are what turn a taken name into an available one, and that usually means two words instead of one.
Brandable, for names that are invented
Brandable searches for names that are not made of dictionary words. This is where the short, memorable, invented names come from, the ones that read like a company rather than a description.
Choose it when the name needs to carry a brand rather than do any explaining, or when a traditional search has come back empty on a crowded word. An invented name has far more supply, because nobody is competing for a word that did not exist until now.
The tradeoff is real and worth accepting deliberately. Nobody knows what you do from the name, so you spend money and time teaching people. That is a cost an established brand pays happily and a new business should not pay by accident.
Creative, for a description rather than a name
This is the one that surprises people, and the fastest way to explain it is that Creative treats what you type as a brief rather than as a name.
With it on, describe the business instead of guessing at a name. Something like a bakery that delivers on Sundays is what the search expects, and it generates suggestions from that description.
With it on and a domain instead, the suggestions have little to do with what you typed, because the search is not checking your name against anything. It is answering a description and you gave it a domain. That is the confusing result people run into, and the fix is simply to turn Creative off.
Choosing between the three naming styles
The decision comes down to two questions, and they are usually quick to answer.
Does the name need to do any explaining? If yes, look for descriptive words and use Traditional. If the brand will do the explaining, an invented name is available in far greater supply and Brandable is the search for it.
Do you already have a name in mind? If you do, leave Creative off and search the name itself. If you have a business to describe and no idea what to call it, Creative is the one that helps.
The pair are a pair because they answer the same question in opposite ways. Traditional finds a name that describes you. Brandable finds one that nobody has to describe. Choosing neither, and leaving Creative on while typing a domain, is how the search appears broken when it is working exactly as asked.
One thing the styles do not change
None of them affect what a domain costs. The price of a result comes from the extension and from the registry rather than from how you searched, so the same name costs the same whichever style found it.
Extensions are also worth a second look before settling, since the price difference between them is larger than most people expect. Finding a name you can register covers that side, and what a premium name is explains the results priced above the standard rate.