Elemental Spirits are a different set of pets you can summon in addition to the avatars, they are used generally in an emergency or when you want to kill something quickly. They are fast to cast and ideal for summoning when you are being attacked, when levelling up summoning skill or for strategic battle tactics.
The downside is they use up mana very quickly (4-20 mana per tick depending on your level) and have no pet bloodpact commands; the upside is they are powerful and can be used against enemy weaknesses, they cast spells of there own accord and can use acient magic, which is very powerful. If you are going to use Elemental Spirits, always try to use the spirit that will do the most damage to the enemy, for example use Ice Spirit Pact against flying wind mobs.
Elemental Weakness Chart
This chart is is a standard weakness chart for all FFXI elements, and can be used to judge what avatar, spritit or even what skill chain to use on a monster, by simply seeing what its weakness is. For example if you was fighting a water monster, it would be weak to all thunder attacks.
To obtain the Elemental Spirits you get a scroll similar to a spell scroll at level one, either by purchasing them at a merchant or auction house or loot dropped from an enemy.
Spirits have a free mind, and cast spells by them selves. An interesting fact is though, if they buff you, they will wait until the buff has worn off before buffing you again. Most buffs performed are self only, but occasionally the spirit will also buff the party.
The elemental spirits cast spells according to the level of the Summoner, and they have a list that they use to cast spells in the same order. However this order can be broken if an event triggers the spirit to cast another spell, for example if the Summoner has red HP, the spirit may break the list order and heal the Summoner.