"I think Mr. Tanaka's mother is a teacher."
is "da" the plain form of desu in this sentence? I wasn't sure whether it was a part of the sentence structure.
Yeah - だ can be considered the plain form of です when applied to a noun, it functions similarly in expressing a state of being, and inflects in similar ways for the past and such.
You'll also see many occurrences of です though which can't be replaced by だ and which merely serve to make a sentence more polite - particularly those which follow an い-adjective. Those occurrences are also strange in that they can't be inflected:
楽しい - (It) is fun (plain)
楽しいです - (It) is fun (polite)
楽しかった - (It) was fun (plain)
楽しかったです - (It) was fun (polite)
Let me just add to that by saying that the opposite is also true, ie, there cases such as this example where だ must be used. Typically, the polite conjugation must go at the end of a sentence, and within the sentence だ must be used instead of です for state of being.