Which scientist's laws describe planetary orbits as ellipses?

Kepler, and it cost him years of work he did not want to do. He began convinced the orbits must be circles, because circles were perfect and God would not use anything less, and he spent something like six years trying to fit Mars to a circle before giving up. His own account describes the failure in detail.

What forced him was the quality of the data. Tycho Brahe had measured planetary positions to about an arcminute without a telescope, over decades, and Kepler's circular models fitted Mars to within eight arcminutes - which would have been triumphant with anyone else's observations and was unacceptable with Tycho's. He wrote that those eight minutes pointed the way to a complete reformation of astronomy. A sloppier data set would have let him keep the circle.

The three laws are also the bridge to Newton. Elliptical orbits, equal areas swept in equal times, and the relationship between orbital period and distance are exactly what an inverse-square force produces - and Newton later showed they follow from it necessarily. Kepler himself had no idea why they held; he suspected some magnetic influence from the Sun, and he spent much of his life casting horoscopes for money, which he described as the foolish daughter supporting the wise mother.