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Optimistically, better education and medical care mean that child mortality drops and women stop having more than one or two children, because those will survive and are worth investing time and money in, and besides, designer clothes and iPhones mean most people can't afford more than that. Technology gives us clean energy and recycling and fresh water, and everyone lives happily ever after in peace and prosperity.
We've got something like 150 years to sort this out - an eyeblink of time compared to the 200,000 years we've been around, or the 5000-odd years of recorded history.
Pessimistically, various wars, plagues, famines etc. reduce the population of developing countries while developed countries abandon foreign aid and try to feed themselves with expensive solutions like multi-storey urban greenhouses with reprocessed water. Worst case, the global economy collapses taking western society with it, and most people everywhere die, leaving some kind of Mad Max remnant that keeps going with salvaged technology but can't actually build anything from scratch more complex than a log cabin, because technology needs a large educated workforce. Any Mars colonists probably die with them because they need technology just to breathe, and you can only patch spacesuits with duct tape so many times.
Actual magic, real supernatural beings, or nanotechnology from science fiction that lets us process trash into food and hoverboards at minimal cost would let us sidestep all this for a while, but exponential growth would catch up with us in the end (in just a few thousand years or less). Teleportation over astronomical distances for a few cents a kilo, like in Stargate, would let us export excess population to other star systems but I don't believe that's remotely possible in the time we have.
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The population will swing back and forth like a pendulum. Periods of prosperity will foster a huge population that can't support themselves. So there is a die off as populations starve to death. Then as populations fall low enough that there are more resources than people, the population will surge again. Until a stable balance is reached, it may go on like this for many generations.