Relays & Comm. buoys
(NOTE: there are 2 feasible explanations for relays and comm. buoys. Either they are "Go ludicrously fast" machines (interpretation 1), or they're wormholes (interpretation 2).) in this chapter I will be covering interpretation 1.
Comm. buoys (interpretation 1): humongous death rays pretending to be communications systems
A comm. buoy in the mass effect series officially works by projecting a 'mass free corridor' and firing a laser or radio communication down it to a receiver. However, even if the speed of light were 3,000,000 c (the same as in an FTL drive) that would still be roughly 2 weeks for a message to cross the galaxy. In addition, a comm. buoy is a light based system, when the main reason lasers (also a light based system) aren't the primary weapons in mass effect is because of beam dispersion. This means that comm. buoys should have a limited range of perhaps a dozen light years at most, and that's for a phased array 20 km across (no I didn't actually do the math for that, I just guessed).
In addition, if you can transmit information at FTL speeds, you can also transmit sensor pings. The only POSSIBLE excuse for there not being FTL sensors in mass effect is Reaper interference. Comm. buoys also prove that the council has the technological aptitude required to focus and direct mass effect fields at interstellar ranges, to be blunt I'm surprised that the council didn't come up with a system to shoot down Reapers entering a system using black holes (oh wait... indoctrination...). Lastly, assuming that each square meter of phased array is worth 100 kilowatts of laser power (which is a bit pessimistic), a 20 km wide phased array would still be a 31,400 gigawatt laser. If you used that trick abusing the mass effect's interaction with temperature I detailed last chapter you could (at minimum) increase the system's effective firepower by 3,000 (the same intensity as for a mass accelerator). to put this into context, 1 second of exposure from this weapon (sorry, I meant transmitter) is equivalent to 22.5 megatons of TNT (at minimum), which could easily vaporize a Reaper (especially since those are kinetic barriers).
Mass Relays (interpretation 1): the ultimate artillery
A relay operates in a similar way to a comm. buoy, that should be understood. Assuming that they accelerate the ship using them to the equivalent of 3 km/s, which maps to 100,000,000 c (3 light years per second), a relay reduces mass to the degree that c is 100,000,000,000,000 times greater than normal. It makes sense for there to be a catching relay to slow down a ship, since otherwise you'd end up with the mother of all FTL field collapse events. To put this in context, an FTL field collapse event results in the object in question moving in the same direction with a relativistic speed such that its time dilation (Lorentz) factor is equivalent to its previous rate of travel in multiples of (external) c. A civilian ship undergoing FTL field collapse at 3,000 c could realistically reduce an earth-sized planet to a new asteroid belt. A weaponized relay can deliver (at minimum) 30,000 times more energy per shot.
If relativistic mass works the way Rule of Cool says it does (hint: it probably doesn't but who cares), this should be enough for the projectile to crush stars into degenerate matter, yank planets out of orbit on a near miss, and possibly produce that terrible horror of the universe, a black hole moving only infinitesimally slower than light.
As to how the Relays operate, an extremely intense low mass/fast light field is projected as a sort of 'tube' stretching hundreds or thousands of light years. The ship in question is then catapulted forwards through this tube via some mechanism I'm not entirely certain of (probably working by twisting spacetime a bit). The receiving/catching relay then decelerates the ship via the same mechanism before doing something to prevent the temperature problem I detailed last chapter. The catching relay is necessary, as without it when the ship in question exited the low-mass tube it would not only become a potentially star-killing RKV, it would also explode into plasma.