Respawn beacons are the green triangle icons on the map. When you activate one it will give you a protective shield around you until you reactivate it. It is possible to blow these up and if you destroy one, you wont be able to get the trophy for finishing this level. Make sure you check the map to see that all beacons have been activated. If you check the map at the start, you will be able to plan out a route much better. If you are having a hard time with this, you can play a map you find easy, on easy and go and scope out the map so you know where to find all the beacons and the easiest/fastest way to get through them all.
In solo play, activate every respawn beacon in one story location as a Hardcore character