Mass Effect Series – Identifying the Virmire Survivor

"Reaper? A label created by the Protheans to give voice to their destruction. In the end, what they choose to call us is irrelevant. We simply... are."

There is a lot of information about them here -- I'll copy the summary, but the whole page is worth a read (beware of the obvious spoilers of course)

The Reapers, known by the geth as the Old Machines, are a highly advanced machine race of synthetic/organic starships.

The Reapers reside in dark space, the vast, mostly starless space between galaxies. They hibernate there, dormant for thousands of years, before they are given the signal to return.

Their origins are completely unknown. The first Reaper known to have communicated with organic life, referred to by Saren Arterius as Sovereign, claimed that the Reapers have neither beginning nor end.

Even their true name is a mystery; "Reapers" was a name bestowed by the Protheans, and Sovereign stated that whatever the Protheans chose to call them is irrelevant: "we simply... are."

Sovereign also reveals that the Reapers were the original creators of the Citadel and the mass relay network. These massive constructs exist so that any intelligent life in the galaxy would eventually discover them and base their technology upon them – all part of a scheme to harvest the galaxy’s sentient life in a repeating cycle of purges that has continued relentlessly over countless millennia.

The "mass effect" is the (psuedo-)scientific principle behind most of the technologies in the series. For instance, the "mass relays" allow faster than light travel by exploiting the mass effect.

It's also the reason why weapons have unlimited ammunition (what you're actually doing is "venting a heat sink") and the way in which all of the biotic powers work.

It's explained in the introduction to the first game of the series.

In the year 2148, explorers on Mars discovered the remains of an ancient spacefaring civilization. In the decades that followed, these mysterious artifacts revealed startling new technologies, enabling travel to the furthest stars. The basis for this incredible technology was a force that controlled the very fabric of space and time.

They called it the greatest discovery in human history.

The civilizations of the galaxy call it...

Mass Effect

Although it's a spoiler for the first game, by this time in the series we know

that the technology was not invented by the Protheans (as was thought at the time of ME1) but is in fact far older. The galaxy's continual dependency on the mass effect is one advantage that the Reapers tend to exploit.