Gravity acts in a way similar to electromagnetic forces - and is similarly well or poorly understood. Gravity (in general relativity) can be modeled as a curvature of spacetime, where objects travel in "straight" lines in the curved space. (think of two people walking "north" on the Earth, eventually finding themselves "attracted" to each other as they reach the north pole. (using an analogy from the MTW Gravitation book)
Gravity, like electromagnetism can also be described in a quantum mechanical sense as the exchange of particles (gravitons). This model gives exactly the same answers as general relativity in all situations we are currently able to measure. Both spacetime curvature and exchange of gravitons models run into problems at extremely high energies and short distances, but those conditions are far beyond any existing experimental evidence.
Gravity is an extremely weak force, the only reason its significant is that unlike electromagnetism the are no negative masses, so the vast number of atoms in the earth all add their tiny gravitational forces together to produce the gravity we observe. The extreme weakness of gravity is why we can't directly manipulate it except by moving masses around.
In electromagnetism we can observe the "field" that conveys the force - we can directly measure electromagnetic radiation. Recently scientists did the same for gravity, by measuring gravitational radiation. (see LIGO) Because gravity is such a weak force, its an extremely difficult measurement, but the results are now clear.
The original big bang is still mysterious. Lots of smart people thinking about it, but "why" the universe appeared is not known. Its possible that its not even a meaningful question - maybe its like asking "why" there is a north pole, where did it come from? In general though "why" questions can get into deep water of philosophy of science. Mostly science has to be based on making testable theories and then testing them.