First, let's imagine what would happen if the Earth just suddenly stopped.
The Earth's orbital velocity is 30 km/s, which means that if it suddenly stopped, everything on it would still have 30 km/s worth of inertia. The escape velocity of the Earth is about 11 km/s.
In other words, anything on the Earth's leading side would fly off into space, continuing along the Earth's orbital path around the sun. Anything on the trailing side would be pulverized against the Earth. It would be a horrible mess. But even if the Earth slowed gently to a stop, it would still be a horrible mess. Without the outward centripetal force to counteract the inward pull of gravity, the Earth would begin falling towards the sun.
How long would it take? the whole journey would take about 65 days. It would take 41 days to cross the orbit of Venus, and on day 57, we'd cross the orbit of Mercury.
As they days went by, the Earth would get hotter and hotter as it got closer to the sun. A month into the free fall, and the average temperature on Earth would have risen to 50 degrees C. 50 days in and we'd be about 125 C. On the final day, we'd get up to 3,000 C… and then, that would be that.
Of course, this is completely and totally impossible. There's no force that could just stop the Earth in its tracks like that. There is, however, a plausible scenario that might drag the Earth into the sun.
In the far future, the sun will turn into a red giant and expand outward, engulfing the orbits of Mercury and Venus. There's still an argument among astronomers on whether it's going to gobble up Earth as well.
Let's say it does. In that case, the Earth will be inside the atmosphere of the sun, and experience a friction from the solar material as it orbits around, and spiral inward. Of course, at this point you're orbiting inside the sun, so falling into the sun already happened.
There you go. If the Earth happened to stop dead in its orbit, it would take about 65 days to plunge down into the sun, disappearing in a puff of plasma.