Hardware serial uses a real UART built into the mega328 chip and is connected to pin 0 (RX) and pin 1 (TX). Hardware serial can support very high baud rates (over 500000 baud). Software serial is just that, a UART emulated by software. Software serial can use most any digital pins for RX and TX. Software serial is slow. I know that the reference says 115200 baud, but that is theoretical and in the real world 38400 baud is about as fast as you can expect reliable operation.
If you want to use the serial monitor to debug or receive data from the ESP you can not connect the ESP to the hardware serial port. That means (with an UNO) that the ESP must be connected to a software serial port and the ESP baud rate set to match the software serial port baud rate.
Please read the beginners guide that I linked. Many of the questions that new users have are answered there.