Are you talking about the new star trek movie? If so, in that movie, the bad guys would drill a hole into a planet, and then drop some mysterious material into the center of the planet, which would then cause a black hole. It was a far-fetched concept and needed to make use of an unidentified and so-far not-known-to-exist material that the audience just kind of had to accept for the sake of the story. It's science fiction, after all.
To answer your question:
-Since gravity extends infinitely, there is no such thing as "dropping into Earth's gravity". But an object's gravitational force decreases substantially the farther away you go from the object.
-If a planet the size of earth or twice the size of earth were to get really close, there would be some gravitational effects. For instance, the moon, which is much smaller than Earth, has gravitational effects on Earth which effect the ocean tides. If there was a much bigger object that was much closer, it would produce bigger effects, and would likely be problematic.
-If a planet the size of earth or twice the size of earth were to strike earth, it would pretty much be what you'd expect- two big things hitting each other and breaking and causing devastation. There's a hypothesis out there that suggests the Earth's moon was created a long time ago when a small planet struck earth in our early solar system.
-Masses the size and density of earth are nowhere near enough to cause a black hole. It takes something like a collapsing massive star to cause a black hole to form, or maybe it can be reproduced on tiny scales with sufficient density.
^ This.
Also, it should be noted that even our own Sun isn't massive enough to become a Black Hole, should it collapse, which, by the way, it will in 5-6 billion years time.
Rather it will become what is known as a Red Giant, which will extend at least out to the orbits of the inner planets, scorching Earth. Eventually it will cool and shrink becoming a White Dwarf.
It is true that pressure of various kinds can produce Black Holes of less mass, and it certain people were worried that the LHC might produce microscopic Black Holes.
This is indeed possible, but due to their inherent low mass they would disperse almost immediately due to what we call Hawking Radiation.
Whether something becomes a Black Hole or not depends largely upon whether its mass is compressed beyond its Event Horizon, i.e. the radius within which the mass of the object will collapse in on itself and inside which gravity is so high that even light cannot escape.
For instance, the Earth's Event Horizon is at a couple of inches, which means that if you could somehow compress the Earth's mass down to that size, the Earth would become a (probably rather short-lived) Black Hole.
The 'Red Matter' concept used in the Star Trek movie is complete nonsense though, and no such material is know, nor how it could conceivably work within the laws of Physics.
