The War of the Two Brothers is a conflict between two godlike entities who represent the Marvel and DC Universes respectively The brothers are colloquially referred to as Blue (DC) and Red (Marvel).
So far in the war the Brothers have not attacked each other directly, instead the battles are being fought between champions of the DC and Marvel universes respectively.
History of the War
Early history of the brothers
Red is the younger Brother who spent most of his existence trying to step out of his older brothers shadow.
For the longest time both brothers would create one universe with a hero for each. Sometimes they would play with or design more than one world at a time, if a universe was neglected by it's creator for too long it started to decay but the Brothers could always reboot any world they created. Red first started broke away from this pattern with the x-men universe but even that world still revolved around one singular character "Wolverine". Red broke away from that style altogether when he created the MCU, a universe without a singular protagonist. A universe that would continually expand and grow over time.
With the MCU, Red was finally able to step out of his brothers shadow.
Seeing what his little brother had created Blue immediately became jealous and wanted to put his younger brother back in his place..