The iconic starbird crest has been a major symbol in the Star Wars franchise since day one. Not only does it represent the Alliance to Restore the Republic, but it also became the emblem of the New Republic as well as the Resistance. Despite its widespread use, though, the origins of the starbird crest have never been explained in any of the live-action films. Depending on whether one delves into Legends or the new Star Wars canon, there are two distinct histories behind it.
In Star Wars Legends (the non-canonical expanded universe), the starbird was the Marek family crest, adopted by the Rebellion in honor of Galen Marek. The son of two Jedi, Galen was orphaned and subsequently abducted at an early age by Darth Vader. In a plot to overthrow Darth Sidious, Vader secretly broke the Sith's Rule of Two and raised the child as his own apprentice. He also gave Galen the codename "Starkiller." Later on, Sidious found out and Vader betrayed his student. Although his loyalties initially remained with his master, Starkiller eventually turned against Vader and gave his life to save the three founders of the Rebellion: Bail Organa, Garm Bel Iblis and Mon Mothma. The Republic also used a similar insignia during the Great Galactic War against the Sith, thousands of years earlier.
The new Star Wars canon presents a very different story behind the starbird crest. In fact, it actually comprises two different symbols: Saw Gerrera's three-pronged crest and Sabine Wren's starbird. Saw's symbol has its roots in the Onderon Resistance, which he led during the Clone Wars with the help of the Jedi Order. The Onderon rebels succeeded in reclaiming the planet, but the Galactic Empire took over not long after. Saw continued to wear a modified version of the three-pronged crest for years as he fought against the Empire.
As for the Wren Phoenix crest, the Mandalorian graffiti artist designed it as her personal signature. It then came to represent Sabine's group of rebels as she painted it wherever they went. Sabine also used it to vandalize Imperial propaganda posters, encouraging others to join the Rebellion. Later, the symbol was adopted by other rebels such as X-Wing pilots Brace Marko and Farns Monsbee, who displayed the symbol on their helmets. Together with Saw's symbol, these emblems were combined to create the official logo of the Rebel Alliance.
While these two backstories are quite unique, the meaning behind the starbird itself is the same in both cases. The starbird, or phoenix, is a legendary, immortal creature that renews itself inside the heart of a nova. In both Legends and canon, the rebels adopted the symbol in the hopes that like the immortal bird, the Republic would soon be restored. Much like its namesake, the starbird crest has continued to live on throughout various galactic conflicts.