Matter is a particular form of energy, and can be converted to other non-matter forms of energy. Examples of this are nuclear fusion within a star*, fission reactions in a nuclear power plant*, and radioactive decay.
*(particularly, reactions that produce a net positive energy output)
It should be noted that matter and mass are not equivalent. Sure, matter has mass, but so does energy. Matter is just one type of energy. Energy has an associated mass, and that's where the equivalence lies, via [itex] E = mc^2 [/itex]. [Edit: for the learned readers out there, yes I might be a bit overly simplistic with this claim. If you must, assume I'm discussing patches of space containing zero net momentum. My goal here is not to get overly complicated.]
Similarly, matter can be formed by converting other non-matter forms of energy into matter. It just doesn't happen very often anymore. This did occur (according to most accepted models) in the early universe during big bang nucleosynthesis. It can also still happen with reactions that have a net negative energy output such as inverse radioactive decay and fission/fusion products that have a net negative energy output. Outside of big bang nucleosynthesis (which doesn't happen anymore), these reactions are unlikely due to their lower entropy, but they do happen rarely in intermediate nuclear reaction products, particle accelerators, photon-photon interactions in interstellar space, etc.
Using Newtonian mechanics, any such matter-energy transformations are ignored completely. In relativistic mechanics, a more accurate portrayal of our universe, they are not ignored.
Under short enough time-scales[**], energy cannot be created or destroyed, leading to the concept of conservation of energy.
A more interesting question is whether conservation of energy holds over long time scales. Things can start to get pretty tricky, particularly when applied to an expanding universe. Noether's theorem becomes applicable here.
[** Another edit: To throw another wrench into the mix, unsurprisingly in quantum mechanics, energy and time are conjugate variables, meaning the their respective wavefunction amplitudes are Fourier transform pairs, and the product of the standard deviations of these amplitudes are subject to the Heisenburg uncertainty principle. This has the implication that for really, really short time-scales, where the standard deviation of the time-scale is really smaller, the minimum uncertainty of the energy is necessarily larger. But, since my goal is not to get overly-complicated, let's just agree to ignore this facet of the topic for now.]