Diem
Agent of Phaaze
A capture card--usually an Elgato or something--and Audacity for the commentary. Though let's plays are something of a dying breed, taken over by streaming. People like the live nature and interactivity of streaming more than watching a dozen hours or more of a let's play where there's no guarantee there will be any entertainment value in a particular episode. With streaming, it's much more laid back and about interacting or responding to the audience, whereas let's plays are under more pressure to entertain constantly.How do you even record let's plays anyway?
Supercuts are also very popular for both formats. Edit together the most entertaining and exciting parts of the gameplay/commentary, leaving out the boring parts. But that's much more time intensive, so most bigger YouTubers hire other editors to do that for them.