Cover

Hard cover and soft cover
There are two ways of taking cover, hard cover and soft cover.

Hard cover
Hard cover is when character you are using interact with the surface, ex he put up his back against it or crouch behind it. This happens any time when you press space next to a cover or sprint up to it. This give you a small DR and increased weapon stability and accuracy.

Soft cover
Now to explain soft cover, and to understand that some things need to be explained. Then there is something in the game called hit-boxes. Think every unit as a they are have boxes for arms, head chest etc. If a projectile from a weapon hit this box, damage will be taken. The game is design so that if you shot with your weapon the projectile will come out that barrel, this of course sound obvious, but because of that some things that you might haven't thought about happens. Because of all units in the units in the game is right handed, and is holding the weapon there after, they will shoot from the right to the center where the cross-air is aimed. To use softcover you simply walk up to a wall or high cover and just point out the barrel around the corner from the right side. This will make you able to shot at enemy unit while they can't shot you. This is because your center hit-box is behind the wall, this is what the enemy is aiming at but they will just hit the wall. The benefits of soft cover is that, if used right, make it impossible to harm you from the side where the cover are. They can of course still flank you from any other side. Also you will not get the DR and increased weapon stability and accuracy. Also not all power can be fired while using soft cover as they will just hit the cover, this is especially common with grenade powers. Soft cover can also be referred to as "right-hand-advantage". Soft cover is not unique to Mass Effect, this is possible to nearly every shooter game there is. But it's in third-person-shooter game it is usable as in first person your vision will be blocked if standing behind a object.

Link
This is a video showing the use of soft cover: