Tite Kubo's hit manga/anime series Bleach gives a lot of thematic depth not just to its heroes and protagonists but also to its villains. After the "Soul Society" arc ended, Bleach introduced the Arrancars as its primary villains -- unmasked Hollows who have obtained their own zanpakuto and humanoid forms. However, they can't fully erase their former status as Hollows.
In Bleach, only the konso ritual, when performed by a Soul Reaper, can purify a Hollow and allow it to move on to the Soul Society. By contrast, the Arrancars of Hueco Mundo are permanently cursed and hopeless beings, and this is even reflected in their anatomy, including their Hollow mask remains on their heads. These mask fragments are mostly cosmetic, but they might go deeper than that at times.
The Physical Properties Of Hollow Mask Remains
All known Arrancars have at least a small piece of their original Hollow mask attached to their heads after being transformed into an Arrancar, usually via the Hogyoku artifact that Aizen stole from Urahara Kisuke. In most cases, a uniformed Arrancar's mask fragment will loosely resemble the original mask, but with some minor cosmetic changes -- almost like a mutation. An example is the 4th Espada, Ulquiorra Schiffer, whose helmet-like remains actually look somewhat different from his original Hollow mask, although both versions include long horns sticking out from the sides.
These remains also tend to be small, covering less than half of the Arrancart's head or perhaps even less than a quarter of their head or face, such as Grimmjow's set of teeth and jawbone on his right cheek or Yammy Llargo's lower jaw remains. Weak, lower-ranked Arrancars tend to have more remains on their heads, giving them an unrefined, brutish look. An extreme case is the Arrancar named Iceringer, who had only a quarter of his face exposed, while most Arrancars have the opposite ratio.
These Hollow mask remains often reflect the Arrancar's personality or interests and will change shape when a zanpakuto release is performed. In released mode, an Arrancart's mask remains tend to be larger and shaped differently than their sealed form, and their new mask remains might look more similar to their original Hollow masks.
Once again, Ulquiorra is a good example of that, his mask remains morphing into a close likeness to what he had as a Vasto Lorde. Coyote Starrk's remains transform from a lower jawbone to an eye patch with markings on it, and Tier Halibel's remains shrink to expose her face, making her look less imposing despite her vastly boosted power. This is an exception, though -- most Arrancars expand their mask remains, with Abirama Redder being an extreme example. When he released his zanpakuto, Aguila, his mask remains expanded until they assumed an avian shape and covered his entire head.
Some mask remains even allude to an Arrancar's personality or methods. Szayelaporro Grantz is a brutal scientist with glasses-shaped mask remains, for example, and the warlike Nnoitora Gilga has mask remains shaped like an eye patch over his left eye. Those remains merely grow bigger when he releases his zanpakuto, including gaining two curved horns. Then there's Nelliel, the fallen Espada whose helmet-like mask remains might allude to her defensive nature, and Barragan Luisenbarn, whose mask remains resemble a crown atop his head. He was once Hueco Mundo's king, after all.
Why Hollow Mask Remains Symbolize Despair
The overarching theme of all Arrancars is despair -- a theme that is reflected in multiple ways, including in a typical Arrancar's very anatomy. As Nnoitora pointed out, Hollows have no salvation to look forward to, and not even the scheming Aizen Sosuke can change that. So, Arrancars like Nnoitora and Grimmjow fill the void with sheer violence and conquest, but they still can't escape the emptiness inside as a Hollow-turned-Arrancar. Even if the Arrancars try to resemble Soul Reapers with their humanoid bodies, zanpakuto and uniforms, they can't escape the curse of being a Hollow.
An Arrancar's mask fragment may serve as a symbol of desperate futility. Even if that Arrancar tears off most of their mask, they cannot get rid of it all, and they can't stop being a Hollow with a hole for a heart. Those mask fragments are a constant visual reminder of a failed attempt to become something more, and ironically, when Arrancars release their zanpakuto to destroy their enemies, they will resemble their original Hollow selves even more, which all starts with their mask fragment growing bigger when the zanpakuto is released. Perhaps deep down, Aizen found all this very funny, or perhaps pitiable. At least, it's clear he never cared for his Arrancar followers on a personal level, no matter how much he empowered them with his words.