note-not including classes that rely on divine magic.
The term sorcerer generally applies to individuals who are born with magical capabilities. As such, they are, under the vast majority of circumstances, tied to one particular element, energy, or type of magic. Sorcerers manipulate their particular part of the magical specturm as an extention of themselves. This gives them an enormous amount of control over their form of magic, allowing them to preform freeform, spur-of-the-moment tactics, as well as pre-trained techniques, both in and out of combat. It also means their spellcasting is dependent entirely on their own energy.
Although they don't require it, some sorcerers find it helpful to have an external/designated point of focus. As such, amulets, wands, staves, even tattoos on the head/hands/chest are not uncommon.
Some sorcerers are self-taught, but most tend to seek out another, more expierenced sorcerer for training. Such training usually consists of controlling focus, meditation, inner energy control, and excercises to keep the spiritual channels open.
A wizard is someone who has learned the art of magic through study and experimentation, using a combinatin of motions, components, incantations, and rituals tied to particular times and/or places to cast spells. It is noted that a wizard does not require any innate magical talent or abilities, as all the power and substance for their spells comes from an outside source, which the harness using the aforementioned techniques. This grants the wizard's spells an enormous amount consistency and reliability.
However, the nature of such spells mean that altering or re-purposing them can be difficult, if not impossible. Any deviation from the standard ritual will more than likely not result in a different-yet-related effect, but in either a total spell failure, or a completely different, unrelated, possibly catastrophic effect.
Because of the innate complexity of wizard spells, wizards will only be able to cast a small handful of spells from memory. As such, most wizards carry around large volumes of spellbooks, as references for any spells they may require.
Wizards vs Sorcerers
One of the main factors of consideration when comparing wizard magic with sorcery is noting where the spellcaster draws their power from. A sorcerer's power comes from within, and is directly proportional to their connection with their magical element, and the strength of their spirit. With regards to wizards, an experienced sorcerer can have much more firepower and control, and react faster, casting with almost reflex-like speed. A sorcerer, for instance, gifted in ice magic, could construct a figurine or statue out of ice on a mere whim, even if they had never preformed such a feat before. A wizard, on the other hand, would have to find an ice-statue spell, with specifications for height, the type of statue, and the quality of the ice. Finding a book that references and details such a spell could take months, and necessitate traveling throughout several kingdoms, if such a spell existed at all. A wizards fireball spell will always be a spell that blows s--t up in a fiery explosion of death. You can't use it to light your pipe, warm your clothes, start your campfire, or send up a flare.
That being said, since the wizard's power for the spell comes from an external source, as long as they have the components and don't lose their voice, they can cast fireballs from sun up to sun down, and keep going long after a sorcerer would have collapsed from exhaustion. And though wizards will often specialize in a particular element or type of magic, nothing keeps them from casting spells of a different type, as long as they have instruction and resources.
In combat, sorcerers tend to be along the front lines, utilizing their quick spellcasting and active response, and falling back once they've run out of energy. Wizards, on the other hand, tend to use others for protection, and blast enemies from afar, as flipping through a spellbook, however hurriedly, is hardly a desirable action in center of combat.
A mage is a hybrid of the sorcerer and wizard classes, combining a centralized energy with structured spellcasting. A wizard might bind themselves to a particular magic or energy type, increasing their power and control over that type of magic, but limiting their range of the type of magics they use. Conversely, a sorcerer might bind their energy within a more rigid structure of spellcasting rituals, which would limit their overall control over their powers, but make the process more efficient, giving them more staying power.
In an unmodified state, a mage would cast spells as a wizard would, but the resulting effect would be of a single element or magical type. In the case of an ice mage, a fireball spell would come out as a chilling blast spell, a sense intruder spell would become a sense coldness spell, and a universal greeting spell would become "Ice to meet you." (ba-dum, chaaaaa)
Enchanted objects provide mages with better intuitive control, remove the need for components, or provide a localized power source for energies out of a mage's usual domain. Rods, magical armor, and orbs or other artifacts enchanted with particular elements are common among a mage's belongings.
Full wizard description, wizard/sorcerer comparison, and addition of mage description