How to make Sprite not Bounce on Collision (2D)

Answer about Position Iterations helps in general, but if you have a very dynamic character - you can set him property in Rigidbody2d > Collision detection to Continuous instead of Discrete (but it could affect on your performance, because continuous consumes more resources).

There are settings in the physics options in project settings for the minimum overlap required for a collision, lowering this might help.

Also, check the other settings.

(old) Make sure the you use the same custom Physics Material on your character as well as your terrain.

I was having the same problem and I fixed it by putting my movement code in a FixedUpdate as opposed to just an Update.

Go to your PhysicsSettings2D in Edit > Project Settings > Physics 2D. There is a field called Position Iterations... Change it to a higher number (maybe 5 or more). This should stop the bouncing.

I'm not sure if I get this right, but if you want it to have a rigid body and not be affected by physics at all you can check the "Is Kinematic" box in the rigidbody2D feature.