A star is a massive ball of burning gas whose main function is fusing hydrogen into helium. They are formed by huge clouds of gas that eventually come together in sufficient quantities to kick off nuclear reactions.
Planets, in turn, come from the material left around the star after its formation. They form from small bits that clump together into protoplanets which in turn smash together to form the large objects we see today. Most other objects in our solar system—asteroids, dwarf planets, comets, and the like—are leftover building blocks of planets that never quite came together.
“Large objects that are formed from dust clouds around a star” is usually the quickest, dirtiest way to define a planet. Except when it isn’t. Some objects fall in an odd in-between—not big enough to be a star, but too big to be planet. And other planets might never have formed around a star at all.
The In-Between

