For a good portion of bosses you can access them through raids, once you enter there is usually a text from the boss on top of your health bar and then the actual boss comes out with different phases, you can also get accessories from these boss fights. The bosses consist of:
Aizen Raid Boss: For more information please visit the article
Ulquiorra Raid Boss: For more information please visit the article
Kenpachi Boss: For more information please visit the article
Hellverse Ichigo Boss, (No longer accessible): For more information please visit the article
Yuuki Terumi Boss: For more information please visit the article
Yamamoto Raid Boss: For more information please visit the article
Byakuya Raid Boss: For more information please visit the article
Starrk Raid Boss: For more information please visit the article
Bankai Boss: For more information please visit the article
Voltstanding Boss: For more information please visit the article
Twin Soul Boss: For more information please visit the article
Segunda Boss: For more information please visit the article
Orochi "The Shura" Boss: For more information please visit the article