FPGA board comes with a microprocessor?
There are 3 shades of microprocessors in combination with FPGAs:


The microprocessor is part of the FPGA firmware, called softcore (e.g. NanoBlaze or MicroBlaze for Xilinx)
The FPGA chip contains a hardwired microprocessor in addition to programmable FPGA gates (e.g. Zynq-7000 contains an ARM). Such complex chips are called System-on-a-Chip (SoC). 
The board contains an additional separate microprocessor chip more or less connected to the FPGA chip (e.g. the Basys2 from Digilent or many other FPGA boards contain a small microcontroller that is used e.g. for USB interfacing or feeding the FPGA configuration at start-up) 


The manual of your boards should tell you what you have. 
As Dave has pointed out there are many applications where is is very useful to have both.
Some requirements are easier/more efficient to implement


in (sequential) software (e.g. higher levels of a protocol stack; or functionality that is necessary but not used often) on a microprocessor and 
some others in (parallel) hardware on a FPGA (e.g. lower levels of a protocol stack; functionality that is needed all the time or that needs to be very fast).