*Long, boring post alert*
As already stated, there are basic forces in the universe that give rise to electrostatics and mavity etc, this is fundamentally physics. The study of chemistry is much more about (though not exclusively) how these varying collections of sub-atomic particles we call elements interact with each other.
Most of the ideas learnt about structure and bonding, even up to A-level are vastly simplified, and give a rough idea, but often fall down when investigated more thoroughly.
So why do atoms 'stick'? Well once you get down to it, it's all about energy. Atoms adopt structures that minimise their energy as best as possible under certain conditions of temperature, pressure etc. This might be like Salt (sodium chloride), in which positive ions of sodium and negative ions of chlorine adopt a face centered cubic lattice of alternating ions, with electrostatic forces (negative attracts positive, etc) holding them together. This makes the material solid and regular, and is the best way to arrange the atoms to give back energy in bonding and stabilise them.
Another example might be oxygen, O2. Oxygen shares electrons over two atoms, creating a bonding effect between them, based again, on electrostatics, where the negative electrons are attracted to the positive nuclei. This is more energetically favourable than an oxygen atom in typical conditions, and so this is how it exists as you know it. Oxygen isn't really described as sticky, as in normal conditions it is a gas. This is because of the interactions between molecules (O2). What's called Van der waals forces (momentary dipole moments - electrostatics!) hold them together, but only a small amount due to various other factors, such as the polarisibility and general size of the molecule. The van der waals interactions are weak, and so molecules are not held together strongly, therefore a gas is observed.
So how come other small molecules like water or methanol exist as liquids at room temperature? This is down again to the interactions between molecules. Both these substances are what's called 'polar', which introduces more electrostatic interactions between molecules. However, they are additionally stabilised by hydrogen bonding, which accounts for many anomalous properties of water and other substances not normally expected. This is an electrostatic interaction between lone pairs of electrons on oxygen atoms in water and hydrogen atoms in other molecules of water. It makes them 'stick' together more, as energy is minimised when it happens. Water is particularly stabilised by this, hence it having a much higher boiling point than otherwise predicted.
So this is why things 'stick'. Forces between molecules or atoms. E.g. in salt they are stong, and it has a very high melting point compared to a substance like oxygen, which is not a crystal, but diatomic, and held together by van der waals forces, or water, which is a liquid. Of course, there are kinetic effects (diamond exists at room temperature despite being metastable) and the basis of interactions is on the properties of the material itself, and is not easily described by one factor alone, it's different types of electrostatic interactions which arise due to the varying nature and properties of the elements.
If you don't understand any terms there, just look them up on google, most are more clearly explained elsewhere. Although everyone else is going on about sub-atomic particles and fusion, that's not too relevant to the question you asked, as you cant understand their significance unless you first know of the chemistry of the material, which explains why some things stick together in some ways and others dont. Though I may have got you wrong and you're actually wondering why nuclei stick together.
This is probably too much for GD!