Offset Lacrosse Head
An offset lacrosse head is where the sidewalls extend from a slightly lower, or backward, position of the throat. From the side-on view, you'll see a definite step down from the throat to the sidewalls. This offset results in a lower pocket and a lower position of the ball.

The lower overall position of the pocket is designed to give you more feel and control of the ball and provide better ball retention. Offset lacrosse heads do have a higher trajectory than onset heads which is due to the ball releasing earlier. The earlier release is what gives you the quicker passing and shooting.
When offset lacrosse heads launched, there were a couple of issues faceoff players had to be aware of. During the faceoff, an offset head puts the throat further away from the ball while the sidewall is right next to the ball. This can be preferable for some players, but it's a minor difference.
Another issue is when a player with an offset head has the clamp, it is possible to push their head off the ball because their head is further off the ground due to the offset. Again a minor consideration if you're heavy and strong enough to hold the clamp.
With the proliferation of offset lacrosse heads, the faceoff concerns in the end will be immaterial.