Assuming the Earth was non-rotating to an inertial reference frame, it would still rotate relative to the sun because it revolved around the sun, with the sun tracing a path along the equator. A day would be a year long. 6 months of daylight, and 6 months of freezing night. The main feature of the day side of the planet would be a massive cyclonic storm centered on the point closest to the sun, which would stay stationary relative to the sun, meaning to the perspective of a person on he planet's surface the storm would move westward a few miles a day. This storm would cover about a quarter of the near side of the planet. The night side of the planet would feature a massive slowly moving ice cap, where the perpetual night time temperature dropped below the freezing point of water. This ice cap would move at the same rate as the cyclone on the day side, and millennia after millennia would grind down any land on or near the equator.
One interesting feature of a non-rotating Earth would be the position of the oceans. Without the Earth's rotational momentum to push it down toward the equator, the oceans would settle around the poles, leaving al but the deepest parts of the equatorial region dry. The moving ice cap I described above would be built out of water vapor blown from the day side by constant trade winds. Eventually the air temperature would drop below the frost point, and the water would snow down onto the glacier below.
Now, if the Earth was tidally locked, ie did not rotate relative to the sun, then relative to an inertial reference frame it would rotate on its axis once every year. One face would be constantly toward the sun in perpetual day, and one side would be in a constant freezing night. This version of a non-rotating Earth would otherwise be the same as the other one: Cyclonic storm on the day side, ice cap on the night side, and trade winds blowing from day to night. Either Earth could be habitable to a point. Both versions of a non-rotating Earth would be more spherical than the real one. Without its rotational velocity pushing out the equator, gravity would pull the Earth into a more perfect sphere.