An IPO is an initial public offering. The company becomes public when they do the IPO - they are a private company beforehand.
It's the first time a company is raising money from the public. The public can basically be anyone over the age of 18, at least in the United States, who has the ability to open up a brokerage account. It's a very broad term since it encapsulates so many people/companies/institutions.
Before an IPO, a company only has access to capital from very select companies and individuals - these are typically, but not limited to, private equity firms and/or accredited investors.
One important note is that this does not mean a company gives away 100% of it's equity (stock) to the public in an initial public offering. Typically, it's only a percentage.
If the company gave away 100% of its equity, there wouldn't be anything left over for management to own, which would basically render the company worthless since there needs to be an incentive for management to A) manage/work for the company and B) take risks with the human and financial capital it has at its disposal. Because of this, there's typically a concentration of a handsome percentage of the company's overall equity that is owned by "insiders" - management, founders, etc.
You're able to see who owns what percent of a company on platforms/websites like Bloomberg Terminal or Yahoo Finance. As an example, Jeff Bezos owns roughly 12.5-15% of Amazon's holding company, Amazon, Inc., (not sure on the exact amount) which is traded on a public exchange called the NASDAQ (AMZN is Amazon's ticker - a ticker is basically a key that's used to look up a company on a particular exchange. If you use Google, you can quickly get stock quotes by searching "NYSE: ABC" or "NASDAQ: ABC" where ABC is whatever ticker you wish to look up)
In the example above, Mr Bezos along with the other owners of the company (when it was private) have given away, over time, roughly 85% of the company's equity (stock) for the general public to potentially own.
I hope this answer is helpful.