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The main problem with choosing a free software license is that they're a giant pile of legalese rather than a succient explanation of their goals.

MIT, BSD: Do whatever you want with this software, but it's not our responsibility (baseline)

Apache: Same but with extra protections for your trademarks/brand

GPL: Requires derivative software and software to also use the GPL license

LGPL: Requires derivatives to also use the LGPL license, but doesn't consider linking to a library to be a derivative

AGPL: Requires derivatives to also use the AGPL license, but considers communication over the network to be derivative