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