A chemical reaction rearranges the atoms present in its starting materials into new molecular combinations, but it never creates a single new atom or destroys an existing one; every atom present at the start is still present, somewhere, at the end. A correctly written chemical equation has to reflect that conservation exactly, showing precisely the same number of each type of atom on the reactant side as on the product side. Balancing a chemical equation means finding the specific whole-number coefficients placed in front of each formula that make that atom count actually match on both sides.
Conservation of atoms is a hard physical constraint, not just a notational convention
The rule that atoms are neither created nor destroyed in an ordinary chemical reaction reflects a genuine, hard physical constraint on what a reaction can actually do, not merely a bookkeeping convention chemists have chosen to follow. A chemical reaction only rearranges which other atoms a given atom is bonded to, breaking some existing bonds and forming new ones, without altering the atoms themselves, which is exactly why the same total number of each element has to appear on both sides of a genuinely correct chemical equation.
Balancing means adjusting coefficients, never the actual chemical formulas themselves
Balancing an equation involves adjusting the whole-number coefficients placed in front of each chemical formula, how many molecules of each substance are actually involved, never altering a formula's own internal subscripts, since changing a formula's subscripts would describe an entirely different substance rather than simply describing more or less of the same one. Finding the correct set of coefficients that makes every element's count match on both sides is exactly the specific mathematical puzzle balancing a chemical equation actually is, and it's precisely this balanced equation that then supplies the fixed mole ratios stoichiometric calculations rely on to predict how much product a reaction will actually yield.
What we're still unsure about
That ordinary chemical reactions conserve the total number of each type of atom, and that a correctly balanced equation has to reflect that conservation exactly through its coefficients, is extremely well established, among the most thoroughly confirmed principles in the whole of chemistry. What's more genuinely a matter of practical technique is exactly how to efficiently find the correct balancing coefficients for a genuinely complicated reaction involving many different elements and compounds at once, since simple trial and error can become impractical for complex equations, and chemists continue to use and teach a range of systematic techniques, algebraic methods among them, for reliably finding correct coefficients even for reactions too complex to balance by straightforward inspection alone.
This sits inside Chemical Reactions & Equation Balancing, one of eight topics in General Chemistry, one of six domains in Chemistry, one of seventeen subjects the app can quiz you on.