The Wikipedia article referenced below is a decent introduction to Logan Square.
Belmont Gardens, mentioned in one answer, is west of Logan Square, not part of it. Palmer Square is one of the sub-neighborhoods of Logan Square.
It's important to note that the city's official boundaries for Logan Square don't correspond to what most people today consider the Logan Square neighborhood. It's generally considered the area west of the Kennedy from Fullerton (2400 N) to Diversey (2800 N) cenetered around the square itself, at Kedzie (3200 W).