Inserting data into table in server B whenever table in server A gets updated
Normally you would use materialized views for sync'ing tables, however your table structures are different which probably rules out this solution. The next solution would be to implement insert/update/delete triggers on your source table. This can cause performance bottlenecks on high volume transactions. The third solution is to implement ETL (extract/transform/load). There are a number of commercial and open source solutions. Informatica, GoldenGate from Oracle are a couple of commercial options, Telend, Jaspersoft ETL, KETL are great open source. http://www.butleranalytics.com/30-etl-tools/ has a list and links to some ETL tools.