For example, if there is a engine failure, it will be displayed on EICAS as a CAS message. As per my knowledge, the pilot will initiate the contact with any ground station nearby for guidance to land safely. Iis there any equipment in the aircraft to send the distress automatically when there is any major failure?
The only system that notifies anyone on the ground is the ACARS that the other answers mentioned. There are all kinds of notifications to the pilots, and usually guidance to help them resolve it. Look up EICAS. There's not a notification to ATC or dispatchers, only maintenance. If the pilots can't resolve the problem they will ask for whatever assistance they need.
In a complex aircraft there are lots of false alarms, sensor failures, momentary problems, etc. that the pilots are able to work out without a problem. It wouldn't be helpful to notify people on the ground for all these. They can rarely do much to help anyway. If the pilots need to divert or declare emergency then they will alert etc.
Sometimes they will confer with their dispatcher or maintenance personnel about what they need to do. The ACARS messages can probably help with this. They can discuss with dispatchers if they need to go to a specific terminal where there are personnel that can fix a specific problem and the dispatcher helps them determine if they have fuel to make it there.
The pilot is the one that's makes the decisions, so there's really no advantage to sending a distress signal to the ground since all they could really do is to contact the pilots.
There is nothing formally standardized as far as I know however newer Rolls Royce jet engines (and presumably those from other major manufacturers) do have a continuous inflight data link that reports engine data in real time to their reporting centers. You can read up on it here as well as this discussion with lots more info.
Aerospace Operations Centres are able to track the health of thousands of engines operating worldwide by using onboard sensors and live satellite feeds. Rolls-Royce are world leaders at equipment health management (EHM)
The centers are potentially able to detect certain emergencies remotely based on the parameters reported.
Since you note the pilot making contact presumably with the radio: Pilots can report an emergency without the use of a radio by Squawking 7700 on the transponder as well.
Most commercial aircraft have a system called ACARS that can be used to transmit flight information and messages between the ground and the aircraft. In many cases, the ACARS is configured to transmit flight data automatically (engine performance, airspeed, altitude, etc) that would include error messages/warnings from the various systems onboard.