Disclaimer: I'm still an undergraduate, but I like to read about this stuff a lot, and I like to talk about it. I'd appreciate it if someone who was more knowledgeable could look through this and correct any mistakes, because I just want to know more. :)
Well, there's a couple of ways to answer this depending on what you mean.
For hydrogen, it's as simple as, opposite charges attract. The nucleus is positive charged (due to protons) and the electron is negatively charged. (Keep in mind that positive and negative don't really mean positive and negative, that's just the convention we use thanks to Ben Franklin. Don't think yes or no, or on or off. It just means opposite.
As to why they attract, well, a neutral charge (that comes from the balance of a positive charge and a negative charge) is lower energy, and Entropy favors states of lower energy.
As to why electrons or protons are charged, that gets more complicated and is well out of my knowledge. Some people who are more familiar with quantum mechanics may be able to explain it better than I would ever be able to.
But only to a point. No one really knows why things have charges.
As for all other elements, it's almost exclusively[Citation Needed] in the heart of stars. This process is very well known and you can read up on it here. Basically when you give a proton enough energy, it can get close enough to another proton (both positively charged) that the Coulomb Force (Like charges repel) is overcome by the Nuclear Force (an opposing force that causes protons and neutrons to attract). But Nuclei aren't just protons and electrons. Everything but Hydrogen also has neutrons, which are caused through the Proton-Proton chain, but that's a bit overkill to discuss.
Long story short, 2 hydrogens fuse to become helium. This continues with 3 helium to make carbon. Then Carbon combines in a bunch of ways to create other elements which combine to create others until you reach iron. This only happens in really hot stars. Iron cannot fuse in the same manner because it takes more energy to fuse 2 iron nuclei than you get out and is therefore energetically unfavorable. But, if the star is massive enough, the pressure causes it to collapse and explode out (highly oversimplified explantion) which provides the energy to fuse iron together to form the heavier elements.
Those elements are expelled out and eventually, combine to form other solar systems. If you repeat this many [citation needed] (Does anyone have a rough estimate of how many times are required to get the current composition in our galaxy?) you get to us with all our known elements. Then we come along and throw atoms at each other at higher speeds to invent new elements.
So yeah, that's how elements are created.