The Michaels cue-bid is an overcalling convention for showing 2-suited hands. Cue-bidding opener's minor suit shows both majors, cue-bidding opener's major shows the other major plus an unspecified minor. In either case, the cue-bidder promises at least 5-5 distribution. David Bakhshi's article summarises the convention.
As with any overcall, the possible objectives are threefold:
Different players have different agreements as to the strength of the cue-bidder, but it's assumed here that the hand is either weak [5-10 HCP] or strong [16+ HCP], i.e. intermediate-strength hands are excluded. For the weaker hand, vulnerability is a factor to be taken into account. As with any pre-emptive bid, the cue-bidder's values should be concentrated in his/her suits.
Responder assumes that the cue-bidder holds the weaker variant of hand: it's up to the cue-bidder to bid on with a stronger hand. In this situation, responder knows partner holds both majors.
Assuming RHO has passed:
All other bids show a non-minimum opening hand, and are descriptive of the hand shape:
If opener rebids four of his/her original major, showing more than six losers, responder passes unless his/her own hand has five or fewer losers. Otherwise, he/she proceeds via RKCB or cue-bids as appropriate.
If opener shows a non-minimum hand, responder may sign off in 4-major if the hands don't seem to be fitting well, for example if opener shows shortage in a suit where responder has values, or if responder has shortage opposite opener's trial-bid suit. It's also possible, though unlikely, that responder may pass a 3NT rebid. Otherwise, responder once again proceeds via RKCB or cue-bidding.
In this situation, 2NT by responder is a natural bid. Cue-bidding RHO's suit becomes the Jacoby bid, forcing to three of opener's suit, and a jump raise in opener's suit is pre-emptive. So now, if opener rebids three of his/her suit in response to a cue-bid, this shows a relatively minimum opening hand. Where opener's suit is spades, this may guarantee at least a 5-card suit, which gives responder more scope in finding the most descriptive bid. If the opening suit is hearts, intervention in spades makes things more difficult, but double is now used as a support double promising exactly three of opener's suit.
There's no longer a Jacoby-style bid available, but responder has several options: