Meanings
Facing two equally dangerous situations; caught in a dangerous situation with no way out

Examples
Young Charlie was caught between two fires: blowing the whistle on his friends that broke the glass and get away scot free or taking the blame on himself and be punished, thus becoming the head of the group?
Carlino era preso tra due fuochi: denunciare i suoi amici che avevano rotto il vetro e cavarsela o prendersi la colpa e la relativa punizione, diventando però il capo del gruppo?

When John invited for a few days his mother and his mother-in-law at the same time, the poor man didn't realize he would find himself between two fires
Il povero Giovanni non si era reso conto che ospitando per qualche giorno contemporaneamente madre e suocera si sarebbe trovato tra due fuochi

Origin
A soldier who was exposed to gunfire from two or more sides was said to be “between two fires”. Figuratively speaking, “fire” refers to any danger threatening from all sides simultaneously