What are some modern fast/flexible Intermodule / PCB-to-PCB busses commonly used with FPGAs/SOCs?
That is, buses that are off chip and off board.

Some I can think of off the top of my head are.

Serial:
I2C, SPI, and RS485: very simple but flexibility and speed are limited.
PCI Express: Fast and flexible, but would need an FPGA on each card to interface to it.

Parallel:
ISA/PCI: fast and somewhat flexible (limited # of cards) but older and some level of intelligence is required on each card to support the bus.
Something like an AXI bus seems like it could fill the roll but It seems like it is an on-chip only buss.

 

There are lots of bus designs that have been done, and new ones come along all the time, like USB 3.0.
FPGA's have a variety of input/output signals and controls. In fact, most FPGAs have pins that can be either inputs or outputs, depending on how you program them. So to help answer your question, can you fill us in with some details of what you want to do? Requirements. Speed, bandwidth, serial or parallel, data path width, voltages, inside another system, duty cycle, etc. https://en.m.wikipedia.org/wiki/Bus_(computing)