Alright, it's really simple.
When you die, you either become a good guy (a shinigami) or a bad guy (a hollow).
Either way, you're going to end up with a sword (a zanpakuto). Each zanpakuto has its own name.
When you learn its name if you're a shinigami, you learn something called Shikai (or first release). Shikai is an ability that's unique to your zanpakuto. Usually, it changes the shape of your zanpakuto and gives it special abilities. For example, the ability to shoot energy balls, the ability for it to extend really far, etc.
After a LONG time of training, you can also learn bankai (second release). Bankai is the most powerful release of a zanpakuto, and gives it incredible, almost TOO powerful, powers. They can be anything from super-speed and the ability to launch energy blasts, the ability to summon a humongous giant mechanical beast that copies your movements, etc.
If you're a hollow, you'll eventually want to become what's called an arrancar. Arrancar are hollows that have gained some shinigami powers. Upon learning the name of your zanpakuto as a hollow, you learn what's called a Resureccion. It's similar to a shikai/bankai, and either causes you to change your form and gain powers, or causes your zanpakuto to change form and change powers. It can be anything from you turning into a panther-like creature and gaining energy claws, to your zanpakuto turning into four gigantic crescent scythes.
Also, Arrancar naturally have the ability to shoot various energy blasts. One's called a cero. It's (usually) a red blast of energy that might take a little while to charge up, but causes major damage. Another is called a bala. It takes less time to charge up and moves much faster, but does less damage than a cero.
That's basically it.