The planet's orbit is mainly determined by the gravity between the sun and the planet and the initial state of the planet (by "initial state of the planet" I mean the planet must be in motion. Of course, the gravitation from other planets close to the subject planet could also play a minor role. Because the mass of the sun is dominating, compared to the other planets (for example those planets including Earth in the solar system), the gravitation between the sun and the subject planet pretty much determines the planet's orbit.

The initial state of the planet, I mean the planet must be in motion, otherwise the gravitation alone would pull the planet into the sun. Then the orbit is not going to happen, or last long.Given the motion of the planet, then the gravitation between the sun and the planet keep the motion, which is the orbit. Basically the gravitation and its initial motion determines the orbit. Also, as a reminder, the mass and the distance between the sun and the planet determine the gravitation. All of those factors play a role in setting up the motion and the orbit of the planet around the sun.

The sun's gravity affects the orbits of all of the planets in the same way. It is the reason why planets orbit the sun at all, instead of just coasting off into space. The closer a planet is to the sun, the stronger the sun's gravity becomes, which is why planets close to the sun move faster in their orbits than planets far from the sun, but the physics are the same for any planet.