Is it possible for the earth to be sucked into a black hole? If so, what would happen?
It depends on the black hole, but with most black holes the Earth would be sucked in a little at a time -- and it would put on an X-ray light show. More details below.
First there has to be a black hole. If it's formed by gravitational collapse it must have at least the mass of several Suns, so its gravity will overwhelm that of the Sun and we are pulled out of our orbit. So we freeze to death before all the cool stuff happens. Bummer!
Most black holes are much smaller than Earth, so they cannot consume our planet in one shot. An astronomer on the planet Remulac orbiting the star Vega might see what we see when the black hole at Cygnus X-1 pulls gas from its companion star (https://en.wikipedia.org/wiki/Cygnus_X-1):
The material from the Earth would be pulled towards the hole, spiral inwards towards the hole, and get so hot from the gravitational energy that it emits X-rays. That's what the "X" in "Cygnus X-1" means. Some material would be so accelerated that it gets expelled as jets which the Remulac astronomers could also see.
So while we would be doomed, at least our planet would go down in a blaze of glory.
Black holes do not randomly appear. You require a star at least 3 times the size of our sun in order to form a black hole. There are no stars, apart from our sun, anywhere near our solar system. The closest other star to Earth is still 4 light years away.
Even if there was a massive enough star nearby that formed into a black hole strong enough to actually pull Earth out of it's orbit, we would still be so far from the event horizon that we wouldn't even get sucked into it. What would happen is we would go into orbit around the black hole as part of the Quasar that would form.