The basic elements of the Shinobi Realm are, Fire, Water, Wind, Earth and Lightning. Of course other elements can be developed through the combination of these basic ones. The 5 Great Shinobi Countries are named after one of each of the elements. This is because usually the Shinobi residing in that country use techniques that are aligned with that element, however, this does not always hold true.
The basic elements are caught in a cycle. As shown above. Fire is stronger than Wind. Wind is stronger ithan Lightning. Lightning is stronger than Earth. Earth is stronger than Water. Water is stronger than Fire. When two elemental jutsu collide, this cycle must be taken into account. The jutsu that is stronger over the other is more likely to win.
Breakdown:
When battling an opponent take into heart their techniques and elements to avoid being at a disadvantage.
You will start out with one Element. When you reach the Jounin rank you will be able to choose your second element affinity. After that any other elements you wish to master must be done through raw training.
Basic Elements
Fire (Katon)
Katon takes the form of a fire or flame techniques. Ranging from small sparks or flames to large jets or orbs of flame. Katon is a destructive element, focusing more on offense than defense.
Water (Suiton)
Suiton takes the form of water techniques. Ranging from shields or walls to powerful jets or waves of water. Generally Suiton is equally offensive and defensive.
Wind (Fuuton)
Fuuton are wind based techniques. Ranging from gusts of air to tornadoes or hurricanes. Wind techniques are generally more offensive than defensive, making it very destruction in combat.
Earth (Doton)
Doton are earth based techniques. Mostly composed of defensive techniques, Doton allows users to create shields, walls, or barriers. Of course it can be used to manipulate the ground or to attack from below.
Lightning (Raiton)
Raiton are lightning based techniques. Composed of bolts and currents. Lightning is another destructive element that is used more offensively than defensively.
Other Elements
Ice (Hyouton)
Hyouton are ice based techniques. Ice techniques are created from the use of Water and Wind elements at the same time. They can vary from pllars to walls. Hyouton can be used equally defensive and offensively.
Wood (Mokuton)
Mokuton are wood based techniques. Wood techniques are created from the use of Earth and Water elements at the same time. They can vary from further manipulation of the land causing land slides or avalanches. To make up for Dotons lack of offensive, Mokuton is a powerful way to retaliate for Earth users.
For example, say Kakashi Hatake who has a powerful S Class Lightning jutsu, Chidori, clashed against an Earth Style Shinobi. If the Shinobi were to build up a wall of stone and earth, the Chidori could easily cut through this wall, causing it to crumble. Of course if the wall was much more massive and filled with greater amounts of chakra it could resist the strike, but overall the Chidori would still come out on top.
Examples: Goukakyuu no Jutsu (Grand Fireball), Housenka no Jutsu (Mythical Fire Phoenix)
Examples: Suiryuudan no Jutsu (Water Dragon Projectile), Suirou no Jutsu (Water Imprisonment)
Examples: Dai Kamaitachi (Great Cutting Whirlwind), Rasen Shuriken (Spiral Shuriken)
Examples: Doton Kekai (Earth Barrier), Doryuudan (Earth Dragon Projectile)
Examples: Chidori (Thousand Birds), Gian (False Darkness)
Examples: Kokuryuu Boufuusetsu (Black Dragon Blizzard), Makyou Hyoushou (Demonic Crystal Ice Mirrors)
Examples: Shichuurou no Jutsu (Four Pillar Prison), Moku Bunshin no Jutsu (Wood Clone)