let me take Windows as an example. you can simply unzip the .zip file you downloaed and save it to C:\. Then, you have C:\jakarta-tomcat\......
Go to C:\jakarta-tomcat\bin and run startup.bat. Then, upen your browser and hit http://127.0.0.0:8080/index.html.... aha, you have Tomcat running... enjoy it!
If you work on Linux, things are similar. Really simple.