Burp Suite is the tool most web application testers reach for, because it turns your browser into something you fully control. It works as an intercepting proxy: every request your browser makes passes through Burp, where you can read it, pause it, modify it, and forward it. That visibility and control is the foundation of nearly every web attack technique.
The parts you will use first are the Proxy (to capture and intercept traffic), Repeater (to send a single request repeatedly with tweaks), and Intruder (to automate sending many variations, for example a wordlist). Learn these three and you can perform most of the manual testing in the vulnerability walkthroughs on this site.