Make sure you have JDK7 or later installed before trying these steps:
1. Download 8.0.5 Core binary distribution having filename, apache-tomcat-8.0.5.zip from http://tomcat.apache.org/download-80.cgi
2. Extract in a suitable location (e.g. ~/softwares/). And change directory inside apache-tomcat-8.0.5/bin.
3. Make startup.sh, shutdown.sh and catalina.sh executable (chmod +x filename)
4. Run startup.sh (./startup.sh)
5. Open your browser and goto http://localhost:8080 to check if its running.