An 83(b) election is a letter to the IRS saying: tax me on this stock today, at today's value, even though it has not vested yet. It applies to restricted stock (actual shares that can be forfeited) and to options you exercise before they vest. It does not apply to RSUs at most public companies.
Why would anyone volunteer to pay tax early? Because early is usually cheap. A founder or early employee who receives 100,000 shares worth one cent each files an 83(b) and reports $1,000 of income. Without the election, each vest is taxed at the value on that day. If the shares are worth $5 when the last block vests, that is income of hundreds of thousands of dollars, taxed at ordinary rates, with no cash to show for it.
With the election, later growth is a capital gain, taxed only when you sell, and the long-term holding clock starts at the grant date instead of at each vest.
The rules are strict: the election must be postmarked within 30 days of the grant or early exercise, and there are no extensions. The common mistake is simply missing the window. The other risk is paying tax on shares you later forfeit or that become worthless; the tax paid is not refunded.