Gravity and inertia keep the planets in their orbits. The planets are moving about the Sun, but the Sun is attempting to "pull them in" at the same time. The planets, as they move on their orbital paths, are operating in equilibrium where gravitational attraction is balanced by the inertia of the planet. As a planet continues to try to move at a tangent to its instantaneous position on the curve of its orbital path, the Sun pulls it toward itself. The result is that the planet just ends up staying in orbit about the Sun, and it follows its orbital path because at every instant, the two factors (gravity and inertia) are fighting a draw for control of the planet.
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