It'll probably do nothing on its own but to get that amount of signatures in less than 27 hours is impressive. May's live speech to the nation yesterday evening is undoubtly a big factor. Any leave supporters I've seen on the net today about the petition is bricking themselves!
As to the referendum from 2016 in question, I'll try and clear some things up. That 2016 poll was strictly speaking only an "advisory" referendum as due to Britain's unwritten constitution, Parliament is sovereign and it is not possible to have a binding referendum even if the bill passed to make it so because it can (a) be revoked by MPs the same way it was passed, and (b) a current sitting of Parliament cannot bind any future Parliament to enact something. For example a current Conservative government cannot force a future Labour government, if it won the next general election, to do something that they disagree with, and vice versa.
The result of the 2016 referendum was codified into law as a statutory instrument, which sets out the order for the UK to conduct its exit from the EU, including triggering Article 50 and when she shall leave. This means that by default if the UK & the remaining 27 EU members cannot agree on a Withdrawal Agreement, the UK will leave the EU with no agreement in place. This was going to originally be March 29 but has now been agreed to be pushed back to April 12 at the earliest and May 22 at the latest.
For the UK parliament to formally revoke the Article 50 notice for leaving the EU, neither May or her government can unilaterally do so on their own. It would be for parliament to do so either by revoking the bill or pass a new bill amending the old one. In an emergency Mrs May or whoever the Prime Minister could inform the EU that she is withdrawing Article 50s notice of withdrawal provided she got consensus from her party and other party leaders to agree to have MPs vote on retroactively apply the revocation at the earliest opportunity.
All in all, the whole thing is a mess and right now the UK is standing naked being laughed at by the rest of the world by something of its own making.
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