Ok. Earth is rotating at a speed of about 1100 miles per hour. If our planet suddenly stopped rotating, the atmosphere would still be in motion at that speed. The atmosphere would be moving so fast it would literally sweep the land masses clear of anything not anchored to bedrock, this would mean rocks, soil, trees, buildings, people and animals. All would be swept up into the atmosphere.
If the Earth's rotation slowed down gradually over millions of years, and this is the most likely scenario, it would be a very different story. If the Earth slowed down to one rotation every year, called synchronous rotation, every area on Earth would be in either sunlight or darkness for one year. This would be similar to what the Moon goes through where for two weeks the front side of the Moon is illuminated by the Sun followed by the front side being in darkness for two weeks.
But what if the Earth stopped rotating completely? In that case, one half the Earth would be in daylight for half the year while the other side would be in darkness. The second half of the year it would be reversed. Temperature variations would be far more extreme then they are now. The temperature gradient would affect the wind circulation also. Air would move from the equator to the poles rather then in wind systems parallel to the equator as they are now.
Even stranger would be the change in the Sun's position in the sky. In the above scenario, Sun would just have a seasonal motion up and down the sky towards the south due to the orbit of the Earth and its axial tilt. You would see the elevation of the Sun increase or decrease in the sky just as we now see the elevation of the Sun change from a single point on the Earth due to the Earth's daily rotation.
As an example, say we live at 30 degrees North latitude. In the Summer, at a longitude where the Sun was exactly overhead, it would slide gradually to the horizon as Fall approached, but since the Sun has moved 90 degrees in its orbit, it would now be due west. As Winter approached, you would now be located on the dark side of the Earth. You would have to move to a longitude 180 degrees around the Earth to see the Sun 1/2 way up the sky because in the Winter, the Sun is 50 degrees south of its summer location in the sky.
There would be other effects of the Earth's rotation slowing also. The magnetic field of the Earth is generated by a dynamo effect that involves its rotation. If the Earth stopped rotating, the magnetic field would no longer be regenerated and it would decay away to some low, residual value due to the very small component which is 'fossilized' in its iron-rich rocks. There would be no more 'northern lights' and the Van Allen radiation belts would probably vanish.
Those were disaster scenarios; now, let's look at the physical effects. Let's take a look at the Coriolis Effect, which simply explains why the hurricanes and tornadoes turn as they do. Now, the direction of rotation for these storms are dependent upon the rotation of the Earth. If the Earth spun the other way, they will spin the other way. In the bigger picture, the opposite rotation of the Earth will result in changed weather patterns, meaning new climate zones and a detriment to biodiversity.
The moon is drifting further away from Earth. This is a result of the diminishing gravity that the Earth holds upon the moon. Now, let's say that the Earth's core just simply stopped. The moon will not collide into Earth; instead it's inertia will send it on a tragectory around Earth.
All in all, that incorporates worst case scenarios and more "realistic" scenarios. Taking into account how the Earth's magnetic field has changed multiple times throughout its history, it can be assumed that the world will not end if our core slows down progressively and spins the other way. The bottom line is this: a percentage of us may die, but life will continue to thrive upon Earth.