These are the strong flattish chords that run from the forearm to the bones in the fingers and thumb on the back of the hand and their job is to straighten the wrist/thumb/fingers. These usually become damaged by a cut (or a bite!) across the back side of the fingers/hand/wrist. Some of these can also be caused by stubbing type injuries the finger, most often at the end or middle finger joints.
Some of the more common extensor tendon injuries to the fingers and thumb do not have an open wound but are closed, and are described as Mallet/Boutonniere/EPL)
They will usually present with a cut/wound on the back of the hand/fingers with an inability to straighten the finger/fingers with pain, and the injured finger/fingers will usually lie dropped compared to the uninjured fingers.
These open injuries can be treated with a surgical repair but will require involved splinting and therapy after the surgery to protect the repair, usually for 6 weeks, and regain the strength and movement afterwards which can sometimes take up to 6 months.
If these are only partially cut or torn, or closed injuries they can often be managed with therapy and splinting.