The universe itself appears as the shape of a ball. It is divided into halves: the lower half being the world of the living and the upper half is home to the afterlife as well as the Kais (known as Other World). In between is Hell and at the very bottom is the Demon Realm. Outside of the sphere is the Kaioshin Realm: home to the Gods of Creation (the Supreme Kais).
The Living World (The Universe)
The living world is similar to our own universe and is home to galaxies, stars and planets - most notably the Earth. Earth is located at the edge of the universe. There are not many worlds with intelligent life in the universe according to Jaco which is why the Galactic Patrol only has 38 members.
Each planet has a guardian (often referred to as a god). Earth's guardian is Dende and he resides at the Lookout to observe the Earth.
The universe is divided into four quadrants that are each watched over by the Kais. They each have a planet located in Other World. There are four Kais (North, South, East and West) and they are governed by the Grand Kai. Earth is located in the North quadrant and is governed by King Kai.
The God of Destruction, Beerus, resides in the living world.
At death, a spirit travels to Other World at the Check-In Station and is sent to Heaven or Hell by King Yemma. If a person is especially good, they are able to keep their bodies after death. At one end of the Check-In is Snake Way which leads to North Kais planet.
Above the Check-In and planets of the Kais is the massive planet Heaven. Here is where the good spirits of the living world are sent and are able to keep some of their original form.
Above Heaven is the planet of the Grand Kai.
Hell (or HFIL - Home for Infinite Losers) is located between the two realms. It is governed by demons and is filled with mean, unhappy people.
The Demon Realm is located at the very bottom of the sphere but is technically not part of the living world. It is out of the Supreme Kais jurisdiction. It was once ruled by Dabura.
The Kaioshin Realm is located outside of the sphere containing the universe and Other World. This is home to the Supreme Kais and their sacred world. The Supreme Kais govern both the universe and Other World. There are only two Supreme Kais left as the others were killed or absorbed by Majin Buu.
Dragon Ball Multiverse
The concept of a multiverse was introduced in Dragon Ball Super. It is stated that there are 12 universes within the multiverse. If two universe numbers are added together and the sum equals 13 they are considered "twin" universes and share a lot of similarities. In episode 47 of Dragon Ball Super, Whis explains to Goku and Vegeta that there used to be 18 universes in total, but one day the Omni-King destroyed 6 of them because he was in a bad mood.
The original Dragon Ball universe is the 7th universe.