A person playing Eve-Online would, upon creation of their character, have chosen from four different starting races. These are the Minmatar, the Amarr, the Gallente and the Caldari. This "guide" is really just a basic description of the fighting styles of each; each race has positive and negative aspects about its gameplay.
Note that the weapons used by each race can be used by each of the other races; however, the ships of any given race have "bonuses" to the racial weapons. In other words, Amarr ships will have bonuses to the damage of laser turrets mounted on them, since the Amarrians primarily use lasers. But nothing keeps an Amarr ship from mounting projectile weapons, and some do. Using every ship to its best advantage, however, means that the most damage will be done by mounting lasers on the ship, and so most Amarrians will do that. Of course, an Amarrian could just train to fly Minmatar ships if they preferred projectile weapons--and then, you'd want to check out the Minmatar list of features instead.
Before we begin, if you're interested in learning more about the fictional aspects including background and personality traits of your race, head here:
On to the fighting styles.
Amarr: The Amarr are fictionally an advanced slaveholding race, believing in their race's purity and superiority. In-game this is reflected with their advanced laser weapons systems. There are two different types of lasers--pulse and beam--for long and short range. The lasers must be fitted with crystals--these crystals both focus the beam and can modify the damage. The bonus of this is that an Amarrian will not have to carry around ammunition or stop to reload his weapons. The negatives are that the crystals eventually wear down, and can do so in the middle of battle if the fighter doesn't pay attention. In addition (and this is very important) the lasers use capacitor energy to fire. This means that there is less energy available to run a tank (see the Term List for more), and that if the ship runs out of capacitor energy--whether because the ship used it all, or because an enemy is attacking with an energy neutralizer or Nosferatu (energy drain), the weapons will cease functioning until the cap recharges.
Amarrian ships are generally very tough. They are also the second-fastest in the game, in general; in accordance with their "technological superiority" aspect, the ships tend to be very shiny and sleek, often looking like lumps of pure gold. The Amarrians have one of the best Battleships in the Eve universe, and the rest of their ships can generally hold their own in a fight. In addition, the Punisher is one of the two best frigates in Eve.
Caldari: The Caldari are often called "the Chosen Race" of Eve, not because of any fictional feature but rather because CCP (the game's creators and developers) often appear to favor this race when coding new game elements. In fact, the majority of Eve players are Caldari.
As a basic overview, the Caldari ships are unbeatable when it comes to PvE work--missioning and the like--and are extremely valuable in group support, using ranged firepower and ECM to help turn the tide of battle. In solo PvP, however, they find their jobs a bit harder; Caldari ships are hard to fit with tanks, tackling gear and damage at the same time, due to the fact that they are heavily biased toward shield-tanking and find it hard to find enough mid-slots to fit their ships.
The Caldari use missiles to their full advantage. A Caldari fighter can be seen in battle sitting far back beyond his allies, nearly invulnerable to attack as he launches volley after volley of long-range explosions. Missiles do excellent damage, but tend to have a low rate of fire; this is often made up for, however, by the Caldari ship bonuses. The Caldari have good frigates (the Merlin and Kestrel are both good fighting ships), a great ratting and missioning cruiser (the Caracal), an excellent, some say overpowered Battlecruiser (the Drake), some great electronic-warfare ships and an excellent battleship along with a great electronic warfare battleship.
The Caldari also use hybrid turrets. The Ferox is the ship to most take advantage of this, and is often set up as a long-range gunboat. The Moa, too (a cruiser) also tends to use these guns. The tier 3 battleship Rokh gets bonuses to railguns, long-range hybrid weapons. Hybrid turrets, however, are less common than missiles in the Caldari fighting style.
The negative aspects to using the Caldari race for fighting are few, but include the fact that large amounts of cargo space must be taken up by missiles or ammunition, that once fully-trained as a Caldari fighter, one can only really use missiles or hybrid turrets, and that the missile systems of the Caldari have a low rate-of-fire. Currently, however, the pros of this race far outweigh the cons.
Gallente: The Gallente are being termed "CCP's Second Chosen Race" by some. This race is a race of excellent fighters; their background, fictionally, is of a people who cherish and preserve freedom by whatever means necessary, and are ingenious fighters as a result, with a small variety of warfare tactics available to them. The Gallente tend toward EW (Electronic Warfare) and drone use. Their top-notch battleship, the Dominix, is something to fear coming up against: it will lock, Nos/neut and scramble you, and then release hordes of combat drones to chew your ship to pieces. (If, by the way, you do not know any of these terms, keep the Term List open in another window to help you.)
Gallente drone use is phenomenal. Their ships tend to have large drone bays capable of unleashing several waves of fighting drones; they also use drones for healing ships, electronic warfare and other purposes. Their ships tend to get bonuses for their drones. In addition, Gallente ships use hybrid turrets (much like the Caldari), meaning railguns and blasters, but rely on these much more heavily than the Caldari do. The Gallente have many of the best ships, including the Thorax (cruiser), Myrmidon (battlecruiser), and (as already mentioned) the Dominix battleship along with the Megathron battleship, which can put out massive damage. Gallente ships do massive damage, in general, and usually have strong tanks to boot.
The only real cons to the Gallente include the fact that they need to rely heavily on drones for damage, and that they need to use ammunition (and thus leave some cargo space open, and spare time in battle for reloading). Also, the Gallente are arguably the slowest race in-game, so for those seeking a zoom-and-shoot play style even with larger ships, this race may not be for you.
Minmatar: The Minmatar are arguably the most versatile, but difficult to play, race in Eve. The Minmatar's background is that of a freed slave race, in appearance similar to what would be left of humanity after an apocalypse: ragged, war-torn, but tough, resourceful and determined. This is heavily reflected in both their ships' appearance and fighting style. A Minmatar fighter will find it hard to hold his own against the other races' fighters at first, but once the Minmatar fighter is well-trained skillwise (in other words, with everything trained to a decent level--drones, missiles, weapons, engineering, mechanic, electronics etc.) he can fight anytime, anywhere in any ship class and have a good chance at winning. The Minmatar must train to be able to tank their shields, tank their armor, use drones well, use autocannons and artillery (as projectile weapons are the Minmatar's primary weapon), use missiles, and use afterburners and microwarpdrives. Speed is the Minmatar's friend; some Matari "tanks" consist merely of great transversal velocity (speed at which the ship orbits, making it difficult or even impossible to hit).
Projectile weapons have their pros and cons. The main pro is that the Minmatar fighter pilot can, by selecting his ammunition type, choose what type of damage (electromagnetic, thermic etc.) to shoot at the enemy. The same, of course, applies to rockets and missiles--but the Minmatar use these weapons too. The main con of projectile weapons is that one must carry around a large amount of ammunition to load them with, and there is also the fact that one must pause in battle to reload.
Autocannons are the short-range weapon of the Minmatar. They have an exceedingly fast rate of fire, but don't do a lot of damage per hit, in general, until one gets into tech2 weapons. The long-range Minmatar weapon is the artillery cannon. Artillery has a much slower rate of fire but does excellent damage at longer range. Minmatar are famed for their hit-and-run tactics; their ships can warp in, do massive damage, and warp back out again before they can be hit too badly.
Minmatar ships, because of the lack of specialization of the race, can be anything that they can be fitted to be: armor-tanking missile boats or shield-tanking autocannon gunboats or anything in between. The Minmatar also use rockets and drones. The Matari ships look rather thrown-together and, some say, rusty, but their frigate the Rifter is arguably the game's best tackler and one of the two best PvP frigates, and the Minmatar Heavy Assault Cruiser the Vagabond is nearly unbeatable when flown correctly. The Minmatar cruiser, the Rupture, is a great missioning ship, and the battleship Tempest is a good long-range sniper boat. The main "con" to being Minmatar is that skills must be trained very high before the Matari fighter comes into his own, with training being split into four or five different skill trees.