Can anyone explain the Star Trek astrography (i.e. quadrants)?

All throughout Star Trek one hears of exploration of various quadrants. I assume these are of the Milky Way galaxy. Does anyone know how the travels of the Enterprise relate to our universe in physical space? Was the entire series limited to our galaxy, or did the crew ever visit other galaxies?

In short, what area of the universe was Star Trek exploring?

Answer

Star Trek was almost entirely set in our Milky Way galaxy.

Earth was in the Alpha quadrant along with most of Federation space and the home planets of the Ferengis, Cardassians and Bajorans. One end of the Bajoran wormhole (featured in Deep Space 9) is also in the Alpha quadrant.

Some Federation space was also in the Beta quadrant, together with the Romulan and Klingon homeworlds.

The Gamma quadrant contained the Dominion space and the other end of the Bajoran wormhole.

The Delta quadrant contained the Borg homeworld and is where most of the Voyager series was set.

A pretty clear overview can be found on the map displayed in the main screen of Elite Force:

An image of the four quadrants of the galaxy.

More details on Wikipedia and Memory Alpha.

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