Whew, what a question! This is a very deep question, so I can only give a general overview. Currently, there are two broadly accepted theories in physics for how the universe works: general relativity, which describes how gravity works, and quantum field theory, which explains the other three fundamental forces: the electromagnetic force, the strong nuclear force, and the weak nuclear force. These two theories are incredibly different, so let me explain them one at a time:
General relativity starts off by saying that time is just the same as the three dimensions we're familiar with (up and down, forward and back, left and right): when we combine time with the other three dimensions, we get what we call a four-dimensional spacetime. Now, you might be familiar with the fact that two-dimensional surfaces can bend (think of a trampoline or a rubber sheet); this bending is called the curvature of the sheet. Well, spacetime can be curved as well, and as objects move in this spacetime, their motion is controlled by the shape of the spacetime; if an object is in flat spacetime, it'll move in a straight line, but if an object moves in a curved spacetime, in general it will move along a curved path. The key point is that general relativity says that the presence of matter is what causes spacetime to curve: so, say I put the Earth is a region of space with nothing else in it; then the Earth will just movie in a straight line, because the spacetime around it is flat. But now say I put the Earth around the sun; the sun's mass bends the spacetime around it, causing the Earth to move along a curved path around the sun (if it helps, imagine drawing lines on the surface of a ball; since the ball is curved, all the lines you draw will be curved as well). In other words, "mass tell spacetime how to bend, and spacetime tells matter how to move." This is our best understanding of gravity.
The other forces (like electromagnetism) are explained by a very different theory: quantum field theory. Quantum field theory says that when you have two particles with some kind of charge (say, electric charge), then those particles will exchange another type of particle called bosons. These bosons are referred to as force carriers, because they cause a "force" between the two charges particles, causing them to attract or repel each other. For a concrete example, think of two electrons. Since electrons have an electric charge, they will exchange a boson called the photon, which is the carrier of the electromagnetic force. The process of exchanging photons causes the electrons to repel (the same thing happens for the attracting of an electron and a proton). I know this sounds complicated, but I hope I helped you understand it a little bit better!
So, in short: gravitational attraction occurs because mass bends spacetime, and the curved spacetime causes other masses to move along curves paths in it, making it seems like two masses attract one another. The other three fundamental forces (electromagnetism and the weak and strong nuclear forces) occur via the exchange of bosons between charges particles.
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