Where does Bernouilli's equation come from? How do the units work?

Bernouilli's equation comes from requiring conservation of energy in a flowing fluid. It's derived in your textbook. It says that $P + \frac{1}{2} \rho v^2 + \rho g y = \rm {constant}$, where $\rho$ is the density and $y$ is the height. The second term is related to kinetic energy of the fluid, and the third term is related to gravitational potential energy (near the surface of the Earth): you can see that these terms resemble $\frac{1}{2}m v^2$ and $mgy$ for point particles. At two points along a flow, $P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2$ and the pressure difference $P_1-P_2$ is related to work done.

You can check the units for each term: they will come out to be $N/m^2$ (try it).