The Tonberry is encountered in the "Malicious Goons" side quest, acquired by talking to Madame M at the Wall Market. You'll get this quest pretty far into the game - throughout Chapter 14, where you'll find plenty of other side missions to complete before you attack the Shinra Tower. Before you rush off for your final hurray, here's how to complete this extremely difficult side quest.
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Malicious Goons | Side-Quest Guide
Location: Sector 6 Slums - Wall Market - Talk to Mrs M nearly Route S6-5. She will notify you of a gang attack at Aerith House in the slums of sector 5.
You will find the goons waiting for you in the arena where you met Rude earlier in the game. Instead of fighting thugs, you will have to fight a Tonberry. This creature is a famous monster from the Final Fantasy menagerie, and he's incredibly hard to beat here.
How to beat Tonberry: Avoid his chef's knife attack - he kills in one hit. Dodge and attack after he misses to build his Stagger Gauge faster. You will have a short window before it attacks again after each missed knife lunge.
Before you try to fight Tonberry, stock up on Fallen phoenix and equip Barret with a long range weapon and Raise Materia. That way he can revive your downed teammates and build his ADP without having to get close to Tonberry. You don't need to heal, because Tonberry doesn't do any damage - it kills you, or it doesn't.
Punisher mode can counterattack safely Tonberry's knife. Use it to guard against the Tonberry's teleportation attack. Tonberry is more likely to attack your party leader, so when he teleports, switch to Cloud and keep in Punisher mode. You are guaranteed to survive a sudden knife attack.
Keep Barret away from Tonberry to avoid his killer frozen attack. The Tonberry can stop time in a small area of effect around its body. When he charges up this attack, back out - or swap with Barret and keep shooting while he's out of range. When Tonberry is low on HP, it will stop time in a small area around it. Use Barret to distract Tonberry after casting this spell to save your allies.
Tonberry is only immune to sleep. Other status effects like Poison will work on him - but they're not too effective. Time Materia also helps slow Tonberry or speed up your ADP gain. Unfortunately, there is no easy way to guard against Tonberry's knife attack. It's an instant kill, so I mostly recommend using Cloud - use it to bait Tonberry. If you're in Punisher Mode and on Guard, you're almost always safe.
The trick is just to avoid letting Tonberry kill your entire party. Use Raise when you can, or use Phoenix Downs. Stay alive, take it down slowly and you will win. There is no grand prize for completing this side quest - just one step closer to 100% completion.