Each shinigami has a weapon (zanpakuto) and this is the way they release their own powers, but how do the shinigams acquire these objects?
Different souls have different spiritual pressure (reishi). Through training, souls can improve their use and control of that power (most often within the Gotei 13). Once they gain enough power, they have to determine the name of their zanpakuto. We never see anyone below the rank of officer either name or release their zanpakuto (not counting Ichigo), and this is in keeping with the "training = release" theme through Bleach
A Shinigami can obtain their Zanpakutō by solidifying their Rieryoku into a Sword form. According to tradition, when a Shinigami graduates from the academy they will gain their own unique sword. However the actual process in which this is done has not been shown. What is known about the obtaining of a Zanpakutō was shown through Ichigo Kurosaki. He was trained independently in a perhaps unorthodox manner and gained his Zanpakutō through means of digging deep into his own soul to grab the power he had never had before. Whether this is the method used commonly for Shinigami is also still unknown. When a Shinigami obtains their own Zanpakutō they learn it's name as a way of "activating" it. This activation of power will release the Sword's Shikai, the first released state of any Zanpakutō. When the sword changes into this form, what is actually being seen is not the change of state, but the true form of the Zanpakutō.
If wikipedia is any source, it states, in an article on the list of Shinigami:
All Soul Reapers possess a zanpakutō (斬魄刀, lit. "soul-cutting sword"), a katana which reflects aspects the user's soul and personality. A zanpakutō has a symbiotic connection with its owner, its spiritual embodiment possessing similar traits to its owner and evolving to reflect its Soul Reaper's power. By learning the name of the sword's spirit, and through training, Soul Reapers can unlock more powerful transformations of their zanpakutō.