First off you will have to learn bribe on the sphere Grid. Rikku will probably be the first character who will be able to learn this.
Then in a battle you will have to select that option on input the amount of gil that you would like to try and bribe the monster with. If it is the right amount you can get some really useful items. Below is a list to help you out.
MONSTER GIL REQUIRED ITEM
RECEIVED Achelous 120,000 Healing Spring x16 Adamantoise 1,088,000 Special Sphere x6 Aerouge 4000 Lightning Marble x4 Ahriman 56,000 Farplane Wind x6 Alcyone 8600 Mega Phoenix x2 Anacondaur 116,000 Healing Water x16 Aqual Flan 40,500 Water Gem x15 Bandersnatch 36,000 Dream Powder x20 Barbatos 1,900,000 Teleport Sphere x20 Bashura 340,000 Stamina Tablet x80 Basilisk 40,500 Petrify Grenade x24 Bat Eye 7600 Silence Grenade x12 Behemoth 460,000 Lv. 2 Key Sphere x30 Behemoth King 1,350,000 Three Stars x14 Bite Bug 4000 Poison Fang x2 Black Element 182,000 Black Magic Sphere x2 Blue Element 30,000 Water Gem x9 Bomb 17,000 Bomb Core x16 Buer 4600 Musk x2 Bunyip 8000 Hypello Potion Cave Iguion 11,000 Petrify Grenade x6 Chimera 105,000 Mana Tablet x10 Chimera Brain 196,000 Lv. 4 Key Sphere x2 Coeurl 120,000 Friend Sphere x2 Condor 1900 Smoke Bomb x3 Dark Element 36,000 Return Sphere x2 Dark Flan 256,000 White Magic Sphere x2 Defender 240,000 Stamina Tablet x20 Defender Z 846,000 Designer Wallet x5 Demonolith 900,000 Lv. 3 Key Sphere x40 Dingo 2500 Sleeping Powder x4 Dinonix 2000 Petrify Grenade x2 Dual Horn 37,000 Hi-Potion x60 Epaaj 174,000 Farplane Wind x25 evil Eye 6200 Musk x3 Exoray 148,000 Turbo Ether Flame Flan 30,000 Fire gem x10 Floating Eye 134,000 Gambler's Spirit x10 Funguar 10,800 Turbo Ether x2 Gandarva 2960 Lightning Marble x3 Garm 4800 Sleeping Powder x7 Gemini 720,000 Stamina Tablet x10 Ghost 199,980 Mega Phoenix x38 Gold Element 24,000 Lightning Marble x20 Grat 80,000 Remedy x40 Great Malboro 1,2280,000 Wings to Discovery x8 Grenade 150,000 Shining Gem x12 Grendel 190,000 Mega-Potion x60 Guado Guardian 40,000 Ether x10 Halma 260,000 Supreme Gem x20 Ice Flan 27,000 Ice Gem x9 Iguion 7400 Petrify Grenade x5 Imp 4567 Lv. 1 Key Sphere x4 Ipiria 3600 Petrify Grenade x3 Iron Giant 72,000 Stamina Tonic Killer Bee 2200 Poison Fang Kusariqqu 8900 Silver Hourglass x20 Lamashtu 5500 Silver Hourglass x10 Land Worm 1m600,000 Dark Matter x2 Larva 29,960 Shining Thorn x10 Machea 360,000 Chocobo Wing x60 Maelspike 200,000 Attribute Sphere Mafdet 14,200 Hypello Potion x28 Malboro 540,000 Wings to Discovery x4 Mandragora 620,000 Return Sphere x24 Master Coeurl 260,000 Warp Sphere Master Tonberry 960,000 Pendulum x3 Maze Larva 44,400 Water Gem x14 Mech Defender 174,000 Al Bhed Potion x99 Mech Guard 25,600 Grenade x50
Mech Gunner 45435 Al Bhed Potion x40 Mech Hunter 110,000 Al Bhed Potion x60 Mech Leader 74,000 Door to Tomorrow x2 Mech Scouter 55,000 Door to Tomorrow Melusine 5300 Petrify Grenade x4 Mi'ihen Fang 3200 Sleeping Powder x5 Murussu 11,600 Hypello Potion x24 Mushussu 13,600 Soft Nebiros 14,000 Poison Fang x6 Nidhogg 40,000 Gold Hourglass x12 Ochu 144,000 Remedy x70 Octopus 90,000 Stamina Spring x20 Orge 188,000 Stamina Spring x50 Phylegyas 33,600 Healing Spring x6 Piranha 1000 Water Gem Puroboros 400,000 Shining Gem x36 Ragora 15,600 Remedy x8 Raldo 4800 Hypello Potion x10 Raptor 4000 Petrify Grenade x3 Red Element 9000 Bomb Core x8 Remora 60,000 Water Gem x20 Sahagin 7600 Water Gem Sandragora 255,000 Remedy x99 Sand Wolf 9000 Sleeping Powder x12 Sand Worm 900,000 Winning Formula x15 Shred 39,000 Hypello Potion x50 Simurgh 4000 Smoke Bomb x5 Skoll 20,000 Dream Poweder x12 Snow Flan 12m000 Artic Wind x10 Snow Wolf 8000 Sleeping Powder x11 Spirit 200,000 Twin Stars x10 Splasher 1200 Dragon Scale x12 Swamp Mafdet 17,000 Turbo Ether x16 Thunder Flan 9000 Lightning Marble x8 Tonberry 4545 Amulet x2 Ultima Weapon 140,000 Pendulum x99 Valahna 174,000 X-Potion x60 Varuna 1,120,000 Megalixir x20 Vouivre 5100 Silver Hourglass x10 Water Flan 6300 Water Gem x2 White Element 7800 Artic Wind x7 Wraith 444,440 Farplane Wind x60 Xiphos 54,000 Megalixir Yat-97 74,000 Ether x16 Yat-99 54,000 Ether x10 Yellow Element 6000 Lightning Marble x6 YKT-11 124,000 Elizir x12 YKT-63 8400 Elixir x8 Yowie 18,000 Petrify Grenade x12 Zanurus 164,000 Rename Card x10 Zu 360,000 Skill Sphere x2
