Not having dug into anything but stuff heard/exposed to while play: Uldren was put in the prison because he started conflict with the Reefborn Awoken prior to the events of Forsaken and Petra has been acting Regent for. Like. Ever now. Or something.

So she had the power to imprison the Prince (King?) of the Awoken?

Much of the Lore cards on the Awoken talk about how much Mara distrusts Uldren. If he allies himself with Fallen, you bet Petra's going to jail him up.

This is a very big question. Here is the very beginning of an answer:

• When the Awoken returned to our solar system (yes, returned. From where? From when? That's another question with its own full blown answer) they found the Earth in ruins and the Traveler hovering low over one single city in the making. Many of the 40,000ish Awoken wanted to immediately head to earth and help.
  
• However, Mara wisely sent Uldren on a scouting mission across the system. He found Fallen on Earth and beyond. They also detected the Cabal on Mars and maybe even the Vex. Mara decided they needed to build their numbers and their forces before revealing themselves to a dangerous, hostile solar system.
  
• In order to do that, the Awoken secretly took up residence in our asteroid belt near the Golden Age wrecks near Vesta, Ceres, etc. They stated building factories and habitats and began growing their numbers.
  
• Unfortunately, a sizable group of Awoken still wanted to go help earth. Things became tense and that group broke away. Mara was forced to let them go or risk a full riot. That group's "betrayal" did indeed begin helping the earth with their superior techonolgy and resources... but they also led the Fallen back to the reef. Which lead to the deaths of many Awoken. That's one of the reason Uldren is no fan of the Earth.
  
• Later-ish, the Awoken started encountering Guardians. Probably Human and Exo Guardians at first, but eventually Awoken Guardians! Meaning, some of those Awoken that went to help earth were killed and then found by Ghosts. Mara isn't a fan, but Uldren sees this as a travesty! His focus is on figuring out how to kill Guardians. Later as the City becomes established Uldren covertly and sometimes overly opposes Guardians... sometimes killing them a few times in a row... sometimes doing mean things like moving or burying their Patrol Beacons to make them harder to access, etc.
  
• Uldren's hatred of Guardians and even his own people (such as Petra) intensifies and grows out of control when his sister sacrifices herself for earth in the battle against Oryx.
  
• After that, Uldren takes over the Eliksni House of Kings and also starts being influenced by *something* that poses as his sister, as seen in game.
  
• Uldren leads the House of Kings in attacks upon the Reef thinking Petra a fool for allowing Guardians to help bring order back after the death of Mara Sov. He allows his hatred of Guardians to translate to Petra and even the rest of his people. For those attacks, Petra and Cayde-6 bring Uldren in and lock him in the Prison of Elders... Ya'll know a lot of what happens next...

But once again, I cannot emphasize this enough: There are layers upon layers upon layers bound up in those few bullet points. Including everything from the Traveler's battle with the Darkness, to the creation of the Scorn, to what happened to Variks are all tied up in Uldren's story. It's some of the best writing in all of Destiny, surpassing or equaling even things like the story of The Last Word or the story of the Hive as told in the Book of Sorrows. My abbreviated answers doesn't do all of this justice...

On a whole I like that some of these things are teased and that we don't get everything explained to us. However, there are some things that I feel should be explained, one of the biggest ones was why Uldren was in prison and why he felt compelled to shoot one of the vanguard.

I have not finished the story, but while the game is fun the story is pretty shit so far. The lore may indeed be really cool, but if you mean to tell me that basic character motivations that give the dramatic conflicts clarity are not presented to the audience even by the end, then it's even worse than I thought.

I see not even a basic understanding of story function and dramatic principles at play here. That's as non hyperbolic as I can put it. I'd be happy to explain all in detail when I finish the story in a post nobody at Bungie will ever read.

*** FORSAKEN SPOILERS ***

As someone else pointed out (was it you, Cheapley?), there’s a pretty big disconnect between what we’re told is happening via the cutscenes (our guardian goes off alone on a quest for revenge) vs what we actually experience (guardians EVERYWHERE, lol). There was definitely room to setup some greater consistency there. As far as the driving plot of the main campaign, I thought it was functional and serviceable (I always knew why I was where I was, and what my goal was). This is actually a fair step up from some of the previous story telling in the franchise, IMO.

If we’re getting down to the motivation of the characters, then I think in some cases it’s intentionally vague/mysterious/misleading (Uldren), other cases I think it’s left unexplored because it doesn’t really matter (the Barons). Uldren’s side of the story is clearly part of what’s being dealt with in the Dreaming City. What I’m curious about is the vanguard. Over the course of D2, they’ve set up a fair bit of tension between Zavala and Ikora. The former’s defensive and conservative priorities are somewhat at odds with the laters renewed dedication to action and boldness. My hope is that they’re actually going somewhere with this. As of right now, Zavala is looking more and more like an old cowardly grump who just shows up to say “no” to everyone while our everyone else (mostly our guardian) does everything. They need to either tell a story where Zavala proves to be right, allowing he and Ikora to achieve and recognize a sort of balance between them, or the rift needs to have actual consequences (perhaps leading to division within the vanguard?).

Being Destiny, I’m tempted to think all this boils down to 1-note “character development”; Zavala is the protective one, Ikora is aggressive, and Cayde is the funny one. That’s all they’ll ever be, and that’s what we’ll get from them until the end of the franchise (we’ll, not from Cayde ;p. #toosoon?). But I’m hoping this isn’t the case. There’s an opportunity to use these characters in interesting ways.

I think it will be interesting to see what happens when we get our cayde replacement. Will he be more like Zavala or Ikora? Will that push the tide to one side? My guess is the person will lean more towards Ikora in which case there might very well be a split in the vangaurd. However, you already find out that Ikora is trying to be responsible by taking a bullet to keep the vanguard in one piece because that's what people need. I think Ikora is just as responsible as Zavala, she is just far more impulsive.

I certainly get the sense that both Zavala and Ikora are equally concerned about responsibility, they're just opposite sides of the same coin.

For Zavala, the continuing survival of Humantiy is his responsibility (in his mind). The events of the Red War have in turn made him somewhat single-minded in his priorities. He refuses to do anything that could jeopardize the safety of the City, regardless of what other gains may be made. On the flip side, Ikora has internalized her responsibility in a very personal way. "The traveler has given me one life to live...". That line, I think, is very representative of her frame of mind right now. Even though she has the light back, she sees it as her personal responsibility to act in whatever way is right. To Zavala, right is whatever is safest for the city. Ikora is thinking bigger than that, which is arguably a higher moral point to stand on, but is certainly more dangerous. And with humanity right on the edge of extinction, its tough to blame Zavala for his cautious approach.

There's potential here, because this is the eternal division and tension between the more open and adventurous types and the more cautious and conservative types. History proves time and again that both approaches have their advantages and their down sides, and finding the proper balance is key to progress and prosperity. But I can't help feeling like its a stretch to think that Destiny's narrative will ever go anywhere near this stuff. We're at the start of year 5 now, and we're just barely, possibly, seeing the first hints of the Vanguard characters being more that hollow, cliche, cardboard cutouts. Still fun to think about, though :)

I honestly don't think they really want us thinking about Zavala this way. They're just using this to manufacture dramatic conflict between NPCs that we resolve with our actions. They've just designated Zavala as the person who feels that maintaining the status quo is the best that can be expected-- a status quo the player breaks by acting.

The problem is that I basically agreed with Zavala, given the information on hand. The real threat in the story wasn't Uldren, it was the person they were working for. That wasn't the thesis Zavala was responding to, though, and Petra never interacts with Zavala. She just recruits Cayde to clean up her mess without actually telling anyone what's really going on, or how her own actions precipitated the crisis (at least, not until it's over) and then Zavala and Ikora fight over whether or not it's worth taking the risk to get revenge on Uldren.

Zavala concludes, rightly I think, that it isn't. It IS worth it to figure out what's going on in the Dreaming City that poses a bigger threat, but that's not the argument Ikora is making, and there's actually no hint that Zavala knows or is ever told anything about it.

If only our guardian could speak more than a few words, we could have told him...

Normal humans don’t know about guardians.

I distinctly remember a normal person in the tower commenting in a random voice line that many people in the city don’t even believe guardians are real.