A spell is an arcane or eldritch feat that allows the caster to weave mana into a supernatural effect. All spells are either at-will (also called 'cantrips'), or ranked feats.
Wizards and Sorcerers both cast spells, but they use them very differently. A wizard casts spells through arduous study and meticulous crafting of sigils, runes, and material components, designed to guide the flow of mana precisely to produce a deliberately engineered effect. A sorcerer, on the other hand, simply wills an effect to occur, and her body intuitively channels the mana in the appropriate patterns to cause that spell to happen.
Regardless of the style in which it is cast, each spell is a very specific effect, designed to precisely manipulate mana in a certain way.
While spells manipulate mana to affect reality, metamagic manipulates mana to affect other spells. A metamagic feat is not the same as a spell - wizards can perform them without preparation, and they are always learned as per-encounter feats rather than daily feats. When you perform a metamagic feat, you typically use it to alter the effect of a spell that you just cast, or a spell that you are about to cast. Some metamagic feats are stances, which allow you to alter the effects of all spells that you cast while in that stance.
All spells require material components to channel their energies. A material component can be an implement, such as a wand, staff or orb, or a temporary object that will be expended when the spell is cast.