Beskar is lightsaber-resistant metal that's only found on Mandalore. The Mandalorians use it to make their armor out of. Boba didn't use it as he wasn't a real Mandalorian, and in addition, didn't have the funds when he first started bounty hunting. He didn't use his father's armor because he was ten at the time, and he had no way to transport the armor that was far too big for him and probably rather heavy. Remember, he had to fight his way out of the arena using only his dad's blaster pistols and his own skill. He made it to someone from the Seperatists who recognized him, and helped him bury his father. Boba felt that it would be dishonourable to take his father's armor, in addition to impractical (it wouldn't have fit without massive resizing). He did, however, replicate it with his own duraplast armor (not durasteel), several suits of which he went through in his lifetime. However, by the time of LOTF, his armor was durasteel, as he was using parts from Jodo Kast's armor, which was durasteel instead of duraplast.