In terms of their structures (primary to tertiary) and locations?
Why do they have these commonalities? Or are any of these commonalities critical to their functions?
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Transcription factors are a very heterogeneous group of proteins. Their two main objectives are to bind DNA & to recruit polymerase to that site. There are a of ways for them to accomplish these tasks.
To bind DNA, all they need is a pocket of positive charge, which are found in several structural motifs across many proteins. The wikipedia page actually provides a good sample of this diversity: https://en.wikipedia.org/wiki/Transcription_factor#Structural
The other co-factors that transcription factors recruit to get polymerase to the DNA strand also vary pretty widely. They link together through structurally complementary pockets of opposite charge, but beyond that, the details get pretty messy and there are a lot of details. Even reviews of transcription factors tend to focus on a specific class. If you want more than that very simplified explanation, I would actually recommend reading some reviews on the subject. Here are a couple: