"I think Mr. Tanaka's mother is a teacher."

Translation: 田中さんのお母さんは先生だと思います。

[Samuel_Davies]

is "da" the plain form of desu in this sentence? I wasn't sure whether it was a part of the sentence structure.

[CaleGibbard]

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)

[TomsAquino4]

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.

[Keith_APP]

It is.