Overall, the p-n crystal is neutral. If it wasn't we would immediately feel it. In fact, it is a fun exercise to calculate how many electrons would be needed to generate a substantial force equivalent to, say, the weight of a car. The answer is surprisingly not very many. This is because the electrostatic force is relatively strong. I am mentioning this to show that the substance would probably be not stable if it had even a small imbalance in the number of positive and negative carriers.
Now, when people talk about P or N doping they mean the following. Each atom of P or N type material is also neutral. However, when placed near each other they cause interesting things to happen. Namely, one of the outer electrons of the N atom "feels" that P atom has a "hole" available (more precise terminology would be a "quantum state"). Due to quantum mechanical effects, there is a non-zero possibility that the electron will transition to that hole. Once it transitions, the N now has one fewer electron and P has one extra electron. Now we have a bit of imbalance: N - slightly positive and P - slightly negative. This is precisely how the PN junction is formed.
when we go about biasing the junction, I.e, applying voltage or difference of potentials to it, we simply encourage certain quantum transitions. Forward bias means that we encourage the electrons to travel from N region towards P region. Of course, on the way between the two, they have to overcome the "natural" barrier that formed as described above. This barrier is the typical 0.1-0.8V potential drop of a forward-biased semiconducting device.