The brainy types of companions are those that possess a higher level of intelligence or more knowledge than the average companion. This gives them a greater capacity for understanding the Doctor's strange world, but initially it can be a point of contention. Take Ian Chesterton, for example. As a science teacher, he had the basics for understanding the Doctor's claims, yet his understanding of science in general was not as complete as he thought it was. Consequently, he ridiculed the Doctor's statements as those of a delusional old man when the fact of the matter was that Ian didn't think that anything not described in a textbook could be true. He was solid and intelligent, but his intelligence caused friction by the Doctor's standards, Ian was little more than a child, and Ian resented having his intelligence belittled. In time the Doctor and Ian learned to work together, the Doctor gradually seeing that Ian was worthy of respect and Ian finding that there was more to science in the universe than he had learned about on Earth. Adric was also a brain-type companion in addition to being a student (it's one of those cross-categorisations I mentioned earlier). Although he was a student as well, the knowledge he already possessed placed him in the brain category, too. He differed from Ian, though, in that his knowledge was already more comprehensive, and he wasn't as intimidated or startled when faced with all the different wonders of the universe. Adric could even understand at least the basics of what made the TARDIS bigger on the inside, something not many companions have been able to achieve. I'm not entirely certain how (or if) Danny will fit into this category. That he is intelligent I don't doubt; you have to be smart to be a teacher. However, it remains to be seen if intelligence becomes one of his defining traits.