The burning of a metal is essentially the same as burning of any other material.

Burning is simply two atoms or molecules combining and releasing energy. This is exactly what happens when a metal burns - the atoms of the metal form bonds with the atoms of the other species and release energy in the form of light and heat. For example, the burning of magnesium and oxygen is given by the chemical equation:
2 * Mg + O₂ -> 2 * MgO.

The products of burning metals are ionic compounds, meaning that the reaction has resulted in electrons being removed from the metal atoms and transferred to O atoms. For the magnesium example, Mg⁽²⁺⁾ and O ^(2-) ions are produced (numbers in parentheses are charge values). The opposite signs of the charges on these ions hold them together, similar to the attraction between opposite poles of magnets. This is not the case for all "burning" reactions though. As one example, burning of hydrogen (2 * H₂ + O₂ -> 2 * H₂O) produces a compound with covalent bonds, meaning electrons are shared between the hydrogen and oxygen in the H₂O. (More details on the differences in these bonds here ).

Typically the word "burning" means that the reaction involves oxygen in the form O₂. This is not a requirement though; other molecules can provide the oxygen, such as in the reaction of Mg with CO₂ (watch the video, it is great!), or iron oxide (rust) providing the oxygen during reaction with aluminum (a thermite reaction). Depending on how one takes "burning", the reaction may not even require oxygen. When hot enough, magnesium will react with nitrogen to form Mg₃ N₂. (Recall that the atmosphere is ~78% N₂. Thus, burning of magnesium in air will produce this compound as well as the oxide.)

[Only semi-related, but perhaps interesting] The reason a metal can burn is the same as the reason any other chemical reaction takes place: doing so rearranges the electrons such that the energy of the system is lowered. Burning of metals produces very high temperatures, meaning that a large amount of energy is released. This indicates that the energy of the metal/oxygen system can be greatly reduced by the two reacting. Given that there is such a large reduction in energy from burning a metal, one might then ask why metals do not ignite easily. As explained in detail here, this is a combination of:
(i) inability of oxygen to reach most of the metal atoms (atoms of the metal don't separate from the bulk as readily as other fuels);
(ii) the energy to break bonds between metal atoms is high (this is part of the cause of point (i) as well), so higher temperatures are needed to start the reaction than in burning of other materials; and
(iii) metals have high thermal conductivity, so any heat that is added is readily spread throughout the object, thereby preventing sufficient buildup of temperature to cause ignition (see point (ii)).

The metal atoms combine with a nonmetal (usually oxygen, since that's what's in the air) to create a salt.

To make a salt, the metal atoms give up some electrons to the nonmetal. This means that the nonmetal atoms fill their electron shells while the metal atoms are stripped of their outer electron shells. This releases energy, which is why salts are stable. The energy is given off as heat.

Note that the common salt on your table is made of the metal sodium and the nonmetal chlorine. The Earth's atmosphere does not contain much chlorine, so burning a metal like sodium in Earth's atmosphere will not create table salt (it will create sodium oxide instead of sodium chloride). To make table salt by burning sodium, you would need to burn the sodium in a chlorine atmosphere instead of an oxygen atmosphere.