You are entirely correct!! :P
*Desu*, the word which corresponds to the verb "to be" in English, and *Da* mean exactly the same thing. *Desu* is the formal version, while *Da* is more colloquial and familiar.
*Da* can sound quite rough and abrupt at times, and is used mostly by male speakers. As a matter of custom, Japanese women tend to speak more politely - even in quite informal situations, you'll hear women use
*"_something_-desu"* rather than *"_something_-da"*.
So, *Da* and *Desu* differ ONLY in the level of politeness, not meaning or tense. It can be used interchangeably.