I opened a free tier account in AWS EC2 and created a Linux instance.
Do I need to pay to Amazon for installing and using Tomcat or is it free for an year?
Unless otherwise stated, EC2 is billed only by the computing power you use (storage, ram, processor, network, etc).
You mentioned Tomcat, which as many other free software offered in linux distributions, can be freely installed regardless of where you run it.
Make sure you review and understand the free tier restrictions before continuing running the instance you've launched.