PDA accepting all words not of the form $b^na^n$
The PDA has two main states:

No $a$ has been encountered.
Some $a$ has been encountered.

While in the first state, it pushes $B$ onto the stack for any $b$ it encounters. It moves to the second state when encountering an $a$, at which point it removes $B$ from the stack for any $a$ it encounters (including the first $a$). If it ever realizes that it has seen more $a$s than $b$s, or if it encounters $b$ again, it can immediately accept (depending on your model, you might have to read the remainder of the input). After finishing reading the entire input, if the stack contains at least one $B$ then it accepts, and otherwise it rejects. You can implement this by requiring the PDA to pop at least one $B$ before accepting.
Formally describing this PDA depends on the exact definition of PDAs in your class, and in any case, it is an exercise which is best for you to do on your own.