Dark space (or intergalactic space) is the name of the physical space between galaxies in the same way as interstellar space is the space between stars within a galaxy. Dark space is completely dark, as it contains little to no stars and makes up almost the entire volume of the universe, where galaxies are only small islands of light. The distance between two neighboring galaxies is in the order of several million light-years, while the diameter of the Milky Way's galactic disc is only about 100,000 light-years.
Dark space isn't completely devoid of celestial objects. Somewhere on the outskirts of the Andromeda galaxy lies the binary brown dwarf 44N81/44N82, as well as an almost barren planet with only calorie-inefficient lichen.
According to Vigil, the Reapers go into long states of hibernation to conserve energy. However, in this state they are vulnerable. By taking refuge in dark space beyond the galaxy's horizon, the Reapers ensure they will not be discovered by accident and destroyed while they wait for their vanguard to open the Citadel mass relay, allowing them to return to the heart of the galaxy.
The last attempt to open the relay to dark space ends in victory for the assembled organic forces and the Reaper vanguard's destruction. Whomever is chosen to be the human representative on the Citadel Council, either Donnel Udina or David Anderson, swears to lead the galaxy against the Reapers and "drive them back into dark space."
Mass Effect 2
Following Commander Shepard's purge or destruction of the Collector Base, the Reaper fleet is seen out in dark space moving towards the Milky Way. The Reaper fleet had apparently already been in transit towards the galaxy since Shepard destroyed Sovereign.
Mass Effect: Andromeda
In 2185, a private venture known as the Andromeda Initiative launched a wave of cryosleep arks towards the Andromeda Galaxy with the stated aim of settling 7 "golden worlds" discovered there. Secret information known only to the highest levels of the project reveal that the migration is partly driven by the impending arrival of the Reapers.
Because of the lack of discharge points in dark space required by prevailing interstellar tech, the Initiative developed or improved several technologies aimed at FTL economy and efficiency such as hydrogen ramscoops and the ODSY Drive.
After over 600 years of crossing dark space and presumably not encountering any Reapers in the big black void, 6 arks successfully complete the intergalactic journey, only to arrive at another galaxy in less than ideal circumstances.