There's a thing in England called the football Pyramid.
Basically the Premiership, Championship, League 1 and League 2 make up what's known as the football League and which league is the higher level is very self explanatory.
Below these league is what's known as non-league and these are made up of semi-pro, ex-pro and armature teams. Basically the league in the non-league work similar to the football league but they get split into regions of the country so you can have two league's i.e. a north and a south league which are on the same level but are two completely different leagues.
The F.A. Cup incorporates all teams in the country, football league and non-league. The League cup only has teams from the Football League.
League cup is that one cup no one (in the premier league) cares about, but is always brought up as an excuse whenever English teams does poorly in european competitions (other countries only have one national cup - in England the FA cup is the most important one).
I wish I had the same drive for soccer as my sister, that way I could actually participate in these discussions.
"Revenge...is like a rolling stone, which, when a man hath forced up a hill, will return upon him with a greater violence, and break those bones whose sinews gave it motion."- Jeremy Taylor