A Zanpakuto is a sword wielded by a Soul Reapers to destroy Hollows. They can appear to be many different styles of swords or bladed weapon.
A Zanpaktou is a mystical sword used by the Soul Reaper in the Gotei 13 from the BLEACH series that was created by Tite Kubo. It first began serialization Weekly Shonen Jump in August 2001. The weapon first appeared in BLEACH Volume 1 CH. 1 "STRAWBERRY & THE SOUL REAPER", and it's first anime appearance is in BLEACH - Episode 1 "Undersea Volcanic Eruption! Drifting to the Fish-Man Island!".
It is a representation of the wielders spirit energy and soul. It's primary use is to destroy the mask of the Hollow. This works in a similar function to the Konso, which allows a dead soul to pass on to the Soul Society. With the Hollow Mask destroyed by the Zanpakuto it washes away only the sins the soul committed after becoming a Hollow.
It's only later revealed that the Zanpakuto is not exclusive to the Soul Reaper. They are also used by the Arrancar Hollows, but the function has a different use.
The first form is an unreleased form which usually looks like an average black sheathed katana for most Soul Reapers, but this can vary. Some have Zanpakuto that are smaller to the form of knives.
Arrancar Sealed Forms
The Zanpakuto of the Arrancar is the sealed embodiment of the Hollow's power and has only two forms. These swords can be in the shape of a basic sword or more elaborate designed blades. Few Arrancar class Hollows even use it as a sword in battle. The released version is closer to the Bankai of the Soul Reaper than the Shikai.
The second form which is the first release known as Shikai can look like a multitude of weapons such as staffs, spears, swords etc. Each having its own personal minor abilities that depend on their fighting style, personality and spirit power of the wielder.
The third form, Bankai or "full release", is the true form of the Zanpaktou and has immense power. For the most part only Lieutenant level and higher Shinigami have obtain this level because the immense time it takes to train to call out the true power of the sword. The powers, form, and limit, greatly depend on the soul, spirit energy, and personality of the wielder. But also depends on the sword itself. As seen with Ichigo Kurosaki, one must become one with their Bankai to finally release it. The shinigami must search deep in themselves and defeat the Bankai as a physical being to be able to use this power.
When Bankai is released the Shinigami usually yells or says "Ban-Kai" to release this power.
The form of the Bankai is not reserved to that only of a sword. The bankai takes its true form which can be anything from a katana to a giant weapon.
Bankai not only changes shape of the zanpakutou, but also gives the shinigami bankai-specific abilities.