The DC universe rebooted back in September 2011, which may be causing some of your confusion. Martian Manhunter used to be a big part of one of the Jusice League incarnations before this reboot. In the current universe, he has some sort of history with the Justice League but was never a full member I think.
In the current universe (the new 52):
Justice League - team of heroes formed during the Darkseid invasion five years ago, basically kicked off an age of superheroes
Justice League of America - the US government (ARGUS)’s attempt to form a counterbalance to the Justice League. Basically they want to have their own Justice League that they can control.
Justice League Dark - a team of magic-related heroes hanging out together for one reason or another. Unaffiliated with either the JL or the JLA
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