Most databases use one-way request-response approach: the client contacts the server and (eventually) expects an answer in return. This, essentially, prevents the server from notifying a client about anything; techniques such as polling, exist, but have their own drawbacks (such as the heavy load on server and network).
On the other hand, you seem to expect a two-ways communication; RethinkDB, already quoted in a comment, is one example of a database which supports this, so it may be a solution for the problem you are trying to solve.
Another solution would be to simply track the changes within your own application, at data access layer. Depending on the language/framework you use, this could be more or less easy to implement, as well as to use technologies such as WebSockets in order to propagate the changes down to the end users.
Finally, if you need to track changes to a table which is used across multiple applications, you may create a web service which will have exclusive rights to modify this table, and make all other applications call this service instead of querying the database directly. On change, the service may use a message queue service to broadcast the message indicating that the data was changed.