Starting with Dragon Ball Z, many of the Dragon Ball franchise's characters have achieved physical transformations that mark just how much stronger they've become as they power up against their opponents. Since many of DBZ's main cast are part of the Saiyan warrior race, these transformations are aptly dubbed Super Saiyans.
However, while many of these Super Saiyan transformations retain much of their users' physical appearance -- save for bright, golden hair standing straight up from the sheer amount of energy -- the Super Saiyan 3 transformation radically changes its users' appearance, with the notable omission of the users' eyebrows, a more pronounced forehead and a more imposing stature.
Goku first unveils the Super Saiyan 3 transformation ahead of his battle against Majin Buu, having learned about the transformation while training in the afterlife for seven years after dying during the Cell Games. Taking several minutes to adequately power up, the sheer energy taken to achieve Super Saiyan 3 is felt around the entire globe and rapidly consumes Goku's remaining time on Earth.
Everyone is stunned by the change in Goku's appearance, with the upgraded Super Saiyan holding a clear advantage over Majin Buu. Gotenks later achieves this transformation himself after Trunks and Goten observe Goku performing the change up close, with the composite warrior matching Goku's altered appearance.
In the anime episode where Goku debuts the transformation, he is seen tapping into what appears to be his more bestial nature, with the Great Ape that channels the full destructive potential of Saiyans visible in his eye as he ascends. However, important to note is that this connection is not made in the same sequence in Akira Toriyama's original manga series.
Instead, in a considerably shorter sequence, Goku completes the transformation across three pages, without a Great Ape in sight. Toriyama's concept art and early designs for Super Saiyan 3, included in the Daizenshu, always had the pronounced brow and lack of eyebrows, while the length of hair varied. Interestingly, one such form appears to include a restored tail for Goku, further lending credence to the primal, Great Ape connection.
Ultimately, Super Saiyan 3 is likely just a method to better differentiate how much more powerful Goku has become in comparison to a Super Saiyan 2, with the drastic change in physical appearance making this all the more clear. The deeper voice, more imposing figure and lack of eyebrows also suggest that Super Saiyan 3 is just a more aggressive form of Super Saiyan -- a transformation largely fueled by anger in most of its forms. This stands in contrast with the more powerful Super Saiyan God and Super Saiyan Blue forms, which not only restore the user's physical appearance -- eyebrows included -- but also appear to provide a calmer mindset.
More aggressive and bestial than its counterparts, Super Saiyan 3 is a form that showcases just how much stronger Goku has become after training in the afterlife for seven years, giving the universe its last, best hope against Majin Buu in its darkest hour; effectively, a monster to beat a monster.
While resurfacing occasionally in Dragon Ball Super, the sheer power consumption required to achieve and maintain the form has led it to be relatively dismissed. Nonetheless, Vegeta's new, divine transformation echoes Super Saiyan 3 right down to its more pronounced brow, longer hair and lack of eyebrows, hinting at both Super Saiyan 3 and Ultra Ego blending the bestial and the divine in a powerful transformation.