This is a Jared solo episode. He tells us everything we need to know about Adric.

He came into the Doctor’s experience in E-Space he was born on the planet Alzarious. His parents passed when he was little from a forest fire so his only family was his brother. Adric was awarded a star-shaped badge for his excellence in mathematics which he almost always wore on his chest in the show. His brother was in a gang called the Outliers. Adric wanted to join but failed and soon after his brother died. 

When the Fourth Doctor came to Alzarius, Adric stowed away on the TARDIS. He was found by The Doctor and Romana and could stay with them. He stayed through the regeneration from the Fourth Doctor to the Fifth Doctor. While he got along well with the Fourth Doctor and Romana, but he annoyed the Fifth Doctor. However, he formed a close bond with companions Tegan (a stewardess from Earth) and Nyssa (an alien from the planet Traken). In an interview the actor who played Adric (Matthew Waterhouse) theorized that the reason the Fifth Doctor didn’t care for his character was that he had many of the same traits as the First Doctor when he was young, which the Fifth Doctor was trying to distance himself from.

His mathematical skills were used a number of times for good (and for bad when he was kidnapped by the master. And in his last serial the Fourth Doctor tried to use Adric’s skills on Logopolis to fix the broken chameleon circuit on the TARDIS.

In Adric’s final serial he complains to the Fifth Doctor that he’s tired of being ignored and teased by The Doctor and he wants to go home. Before that can happen, they discover the Cybermen have infiltrated Earth. It is 2526 and a universe wide conference is being held of all the major leaders on Earth at this time. They are working to form a union to fight and destroy the cybermen. In response, the cybermen are trying to blow up Earth. After helping to find and dismantle the bomb, the Doctor and Adric make up. Then the whole crew take the TARDIS to deep space to stop the Cyber-Leader who is on an Earth owned, military cargo freighter.  With his newfound trust and respect for Adric, he takes him along as they search the ship. The Cybermen take over the freighter and set it to crash into Earth forcing The Doctor to take the Cyber-Leader, and two companions to safety in the TARDIS but Adric insists he be left behind to ensure Tegan and Nyssa’s safety. As the ship moves closer to Earth Adric tries to remove the override controls set by the Cybermen. How is that possible? By solving logic puzzles of course. He is able to unlock one level and the ship (due to the Cybermen’s technology) starts running backwards in time. As the ship is about to crash into Earth, they have not traveled back to around 65 Milllion BC. The ship’s crew leaves on an escape pod but Adric stays behind to try and unlock the last puzzle and save Earth. One last Cybeman, who was left for dead, has enough strength to fire a shot at Adric as he is about to solve the final puzzle. The shot misses Adric and hits the control panel leaving Adric to say wistfully “Now I’ll never know if I was right.” The show cuts to the TARDIS where they watch the freighter crash into Earth. As it turns out, that was the explosion that caused the extinction of the dinosaurs so Earth’s history is intact.

Adric’s death takes a toll on the whole of the TARDIS crew. Tegan makes The Doctor return her to Earth (she rejoins the crew later), Nyssa is deeply affected, and The Doctor blames himself. While Adric is the second companion to die while traveling with The Doctor, the first one was Katrina, clear back with the First Doctor. While we can’t watch all the episodes surrounding Katrina’s death, I would say that the writers made a much bigger point of Adric’s death and how much it weighed on The Doctor. They even had him come back as a ghost to talk with Nyssa and Tegan and the Doctor. This death (as well as the other companions who have died) have weighed on The Doctor ever since, which is why I said he was a companion who’s story arc put him forever in The Doctor’s history.