All about shinigami powers...
Reiryoku, Reiatsu, and Reishi
Effectively, there's three main terms that show up a lot in the Bleach manga when dealing with things related to the shinigami, their powers, and other spiritual-related stuff: reiryoku, reiatsu, and reishi. Of these "reiatsu" is probably most familiar, as it tends to get tossed around most frequently (and at times incorrectly) in the English-speaking fandom, but the other two are at least as fundamental.
So let's start with reiryoku. Literally, it means "spirit power". Reiryoku is the term that describes a shinigami's innate spiritual power. It also can be applied to humans, ghosts, Hollows, or whatever. Basically what this means is, everything with a spirit has some level of spiritual power or reiryoku. A normal soul has very very little. A captain-level shinigami has a lot. In the Bleach universe, only a relatively small percentage of souls have enough reiryoku to become shinigami, but all souls have at least a little. The soul is a spirit, and therefore inherently has at least that minimal degree of spiritual power as a part of its being a soul in the first place.
The next term, reiatsu, means "spirit pressure". Reiatsu describes the actual effect of a shinigami's reiryoku, the almost physical force. It can in many ways be likened to an aura of sorts. A powerful reiatsu can easily paralyze those of sufficiently weaker spiritual power, as well as at the higher levels having an apparent physical force and pressure, creating wind and even crushing the ground underfoot. A shinigami can control their reiatsu to a degree in the form of focusing or suppressing it. Obviously non-shinigami who have sufficiently high spiritual power also have reiatsu, it's again a generic term and by no means limited just to shinigami.
Finally there's reishi. Reishi is "spirit particles", basically it's the matter that all the spiritual beings and things in Bleach are composed of. Ghosts, Hollows, shinigami, all of Soul Society itself, are made up of reishi. Obviously reishi is not normally visible to your average human, though those with sufficient innate spiritual power/awareness can see such things. Reishi can apparently also fuel spiritual energy, as the Quincy derive their power from absorbing the reishi in the surrounding environment and focusing it into their bows.
One interesting thing to note in all this, is that while in Soul Society normally most souls have no need to eat whatsoever, those who possess a high level of spiritual power such as shinigami do in fact get hungry and eat on a regular basis.
Kidou, Zanpakutou, and other Reijutsu
As for the actual application of these basics. Aside from the obvious effects of the reiatsu, there's several other ways in which the shinigami's powers are applied. All of these still stem from the same fundamentals, however, as will be explained for each case.
Kidou is actually perhaps the simplest. Kidou or "demon arts" could be likened to Bleach's equivalent to magic or spells. To use kidou usually involves uttering both an incantation and various actions to invoke the spell, though some more powerful shinigami have been shown to bypass much of this. Using kidou however relies in the end primarily on the shinigami's own innate power and control, however: a shinigami with almost no spiritual power cannot cast a very powerful spell no matter how perfectly they may perform the incantation, and a powerful shinigami who lacks control and focus will have difficulty in making the spell behave as desired. The most basic qualifier for using kidou however is simply one's level of spiritual power: it doesn't matter how skilled you might be if you lack the reiryoku to draw upon in the first place.
Zanpakutou as probably more than well known by now are the shinigami's swords. The name literally means "soul slaying sword", and as the name implies a zanpakutou is a spiritual weapon intended to be used on spirits. It's probably safe to assume that like the shinigami themselves, a zanpakutou is physically comprised of reishi. Zanpakutou are effectively the physical manifestation/focus of a shinigami's powers. All shinigami have one, and a zanpakutou's effectiveness against an opponent is largely decided by the spiritual power and focus of the shinigami wielding it. One of the more important things to note about zanpakutou, however, are the releases. Normally, zanpakutou are "sealed", taking the form of an ordinary sword and scabbard. A shinigami who knows the name of their zanpakutou is generally capable of using shikai, or initial release. This changes the physical form of the zanpakutou, as well as unlocking any ability or power it has. It also increases the shinigami's power and reiatsu, because the seal has been released.
The second release is bankai or full release. Few shinigami are capable of using this release, as it requires years of training both to achieve and master, however it's a prerequisite for achieving the rank of captain. Bankai again changes the physical form of the zanpakutou, this time releasing its full potential. A shinigami using bankai is fighting at a level several times higher than just with shikai. This does obviously also affect things like reiatsu: obviously when the shinigami's power is no longer sealed, the reiatsu also is no longer suppressed. Zanpakutou also have their own identity, meaning that fighting with a zanpakutou is in some ways cooperating with it. They're capable of healing/repairing themselves after being damaged in battle.
There are also other techniques used by the shinigami aside from kidou and the use of zanpakutou. One of the most common examples is shunpo or "flash step". Shunpo is a technique in which a shinigami gathers spiritual energy at their feet, in order to cross large distances in a single step. The limit of this is largely just the strength and skill of the individual shinigami. Also, a few scattered cases of other techniques have been shown in Bleach, largely variations on these basics used by specific individuals. But the fact that such variations exists indicates that shinigami are by no means strictly limited to just these, but have quite a bit of room to adapt and explore new applications.
An aside on the Quincy
The Quincy, being human, don't use kidou. However, a number of Quincy techniques do in fact mirror those of the shinigami. One of the more notable examples is the Quincy hirenkyaku ("flying bamboo-screen legs") technique, which is virtually identical to the shinigami's shunpo. While the Quincy may use different methods and call things by different names, some of the fundamentals apparently remain the same.
All the little details
Now for just some of the miscellaneous little details on things, either expanding on earlier points, or just other information.
First of all and foremost, strictly speaking the shinigami are not human. They started out as human souls and still look human (well, most of them), however they're inherently different. Different to the extent that when one materializes the "spirit threads" of various souls, a shinigami's is actually a different color from the that of a normal soul. Of course because of this, someone with good enough senses for such thing would be capable of detecting if an individual is a shinigami. Similarly, shinigami can sense the presence of Hollows, as opposed to any other spirit. Though of course, as things like reiatsu can be suppressed, there's also ways of hiding oneself from detection.
Shinigami are also inherently spirit-based beings. They're able to inhabit gigai or artificial bodies in order to interact with things in the living world, but they aren't really "alive" in that sense. Even when in a gigai, a shinigami is fully capable of interacting with other spirits such as Hollows and fellow shinigami.
The single main prerequisite for becoming a shinigami is spiritual power. Someone who has virtually no power to speak of simply isn't going to make the cut, period. While on the other hand, someone already capable of manipulating spiritual energy even on a very basic level would probably have no problem getting into the academy for training to become a shinigami. However, reiryoku isn't static. A shinigami can train to become more powerful, and apparently under certain circumstances even someone with little to no powers can gradually gain some.
When in the living world, lieutenant and captain ranked shinigami have their powers restricted. The reason for this is to prevent the immense reiatsu shinigami of such high ranks typical have from adversely affecting everything else in the area.
Aside from killing Hollows, shinigami also send souls on to the afterlife through the process of a Soul Burial. The details are unknown, but the zanpakutou is used in this, so there's obviously some connection with that. Also, it's apparently possible to perform a Soul Burial poorly or incorrectly, but how this works is impossible to say.
Shinigami are able to use jigokuchou or hell butterflies. Within Soul Society these are often used to transmit messages, and shinigami use them to cross between Soul Society and the living world safely. A hell butterfly also appears after a Soul Burial is performed.
Vizard, Arrancar, and Hollows
Here's where the shinigami powers start to get muddled. Both vizard and arrancar use apparently the powers of both shinigami and Hollows, the only fundamental difference being that vizard start out as shinigami, while arrancar are created from Hollows. However, it's not quite so complicated once broken down individually.
To start off with, we need to describe Hollows. A Hollow is a spirit that comes into existence when a soul is not sent on to Soul Society, but instead lingers in the living world, eventually losing its heart and transforming into a monster. Hollows have a hole, typically in the area of the chest, indicating the missing heart, as well as a mask, hiding their former identity as well as covering up the raw instinct they now act on. Hollows vary in power, from being incredibly weak to powerful enough to threaten even the stronger shinigami, and like shinigami, their power stems from their innate spiritual power.
Hollows prey upon other spirits, and especially like those souls with higher levels of spiritual power. It's also mentioned that while an ordinary soul will never feel hungry, one that is about to turn into a Hollow will begin to. What's most notable about this, is that Hollows aren't the only spirits who feel hungry: shinigami need to eat too. Despite coming from far opposite natures, shinigami and Hollows both are subject to some of the same needs as powerful spiritual beings.
So effectively, the difference lies in the nature of the powers, more than anything. The focus of a shinigami's power is the zanpakutou, the symbol of a Hollow is its mask. A vizard is a full shinigami, who then acquires a Hollow's mask, while an arrancar is a Hollow that breaks its mask and seals its power into a zanpakutou. Vizard aren't true Hollows, as they lack holes. Arrancar aren't shinigami, as their zanpakutou is apparently a simple sealing of their powers, as opposed to a focal point drawing them out and augmenting them. All of the above, however, have much in common because of the simple fact that all are extremely powerful spirits, regardless of the type and origins of that power.
