What's the significance of the displayed Robin suit?
Something bad happened to Jason Todd, and Bruce feels very guilty about it.
The premise of The Dark Knight Returns is that for some reason, ten years before the story began, Bruce Wayne hung up the Cape and Cowl. Nobody in the book tells us explicitly what it was that made him do it, but Bruce obliquely suggests that it was because something happened to Jason Todd - Robin in the main continuity:
![Bruce in the abandoned Batcave](https://cdn.statically.io/img/i.sstatic.net/uPa5c.jpg)
Transcript of the relevant bits (emphasis mine):
Bruce: [Narration] I was only six years old when that happened. When I first saw the cave. Huge, empty, silent as a church, as the Bat was waiting. And now the cobwebs grow and the dust thickens in here as it does in me - and he laughs at me, curses me. Calls me a fool. He fills my sleep, he tricks me. Brings me here when the night is long and my will is weak. He struggles relentlessly, hatefully, to be free - I will not let him. I gave my word. For Jason. Never. Never again.
We never find out exactly what happened to Jason, only that it was traumatic, and that Bruce didn't handle it well:
![Gordon and Wayne](https://cdn.statically.io/img/i.sstatic.net/r3ahB.png)
Transcript (emphasis mine):
Gordon: Spoken to Dick lately?
Wayne: Not for seven years, Jim. You know that.
Gordon: Still, huh? I'm damn sorry about that. Especially with what happened to Jason.
Whatever happened, it was serious enough that Bruce doesn't talk about Jason any more - he has a couple of delirious moments later in the book when he thinks he's talking to Dick, but he almost never mentions Jason.
It's commonly believed that Jason has died, and while that's probably the most reasonable theory I'm not sure to what extent it's anything more than a retcon against A Death in the Family. Miller, in any case, was vocally unhappy about the decision to kill Jason in that storyline, saying in a 1991 interview:
"To me the whole killing of Robin thing was probably the ugliest thing I've seen in comics, and the most cynical."
So make of that what you will.
How does it connect with A Death in the Family
As far as I know, no official connection has ever been made; TDKR was always meant to be the definitive final Batman adventure, outside of standard continuity and Comic Book Time1.
Although the influence of TDKR on the Batman franchise as a whole is undeniable - TDKR arguable invented the idea of the gritty superhero2 and informed Batman's character until...well, today - the extent to which it influenced A Death in the Family is probably negligible.
A Death in the Family holds a special place in comic history, and not only because of the infamous comics adage "The only people who stay dead in comics are Bucky, Jason Todd, and Uncle Ben"3; it was also one of the first comics where readers could affect the outcome. As Wikipedia notes, and you can verify for yourself by getting your hands on a copy of Batman #427 (emphasis mine):
At the end of Batman #427, Jason was beaten by the Joker and left to die in an explosion. The inside back cover of the issue listed two 1-900 numbers that readers could call to vote for the character's death or survival. Within the 36-hour period allotted for voting, the poll received 10,614 votes. The verdict in favor of the character's death won by a slim margin of 5,343 votes to 5,271.
Probably the one concrete thing that connects the two stories is the display of Jason's costume in the Batcave, and the more general trend of memorialising former sidekicks:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/WL2hc.jpg)
1 TVTropes link; all hope abandon ye who enter here.
2 There's obviously also a case to be made for Watchmen, and it was definitely a contributing factor. I'm still going to give TDKR credit, partially because TDKR is coloured much darker than most of Watchmen (which gets a lot of mileage out of primary colours), and partially because the first issue of TDKR came out in February of 1986 while Watchmen debuted that September
3 Right, about that...