All metals are reactive by definition, but two groups are considered to be much more highly reactive than the rest. The Alkali Metals group consists of the first row on the left side of the periodic table - lithium, sodium, potassium, rubidium, cesium,and francium. These metals have just one electron in the outermost orbital, called a valence electron. They lose this electron very readily in reaction with other elements, becoming a +1 ion and releasing energy in the process; that's where the reaction comes from. A classic lab demonstration consists of putting a chunk of sodium into water, where it explodes violently. See the youtube link below for a demonstration of this - it's quite amazing!
The next column to the right, the Alkaline Earth Metals, consists of beryllium, magnesium, calcium, strontium, barium, and radium. These metals are not quite as reactive as the first group, but will still react- for instance, magnesium burns very readily. They have 2 valence electrons and will produce a +2 ion when reacting.