There is no real easy way to explain this; there are a lot of moving parts in our political system. Remember that politics is a product of the actions of literally millions of people - so cause and effect can be hard to discern. Our current situation is the result of multiple factors. First of all, you have the congressional makeup right now. In the senate, the democrats have control. In the house, it's the republicans; in particular though, these guys are really more obstinate and conservative republicans than in the past. How did we get here? Over the course of the last decade, the Bush years really shifted our country to the right. Even though his policies became unpopular later, his administration was really good at getting their agenda passed, so by the 2004 election, most congresspeople were neoconservative, establishment republicans. Pretty right wing, but old dogs who knew how to do business. By 2006 though, things had really begun to sour for Bush. It was a big year for democrats, and they took the house and senate. Then 2008 happened and Obama and more dems were swept in. So the dems had a big majority for a while, but they couldn't do anything with it, for two reasons - one is that their party is more diverse/divided on issues then the repubs, and the other was that the elections were more historic events (first black president) and commentaries on the current republicans then votes of approval for a super liberal agenda. So the dems had trouble getting a lot of work done, even though they owned congress. As a reaction to that lack of accomplishment, we had the tea party election of 2010.
The thing about the tea party is that they are kind of the 'last hurrah' of modern(2000's) republicanism. They come in on a big wave in 2010, take control of the house, and are ideologically united - with each other. So they are able to be a real force in congress since they are all on the same page. Their policies aren't really that popular with the country at large though, and demographically, the writing is on the wall for them. One thing a lot of political analysts took away from the 2012 election is that with the number of young women and latinos voting vs old white people, the electorate is going to become more liberal in future elections. Fortunately, it was a bit of luck for conservatives that this all happened in 2010, because every decade the US does a census, and state governments redraw the lines for voting districts. Since the repubs had control of MANY statehouses when this happened, they drew all the maps as favorable to them. This is the most important part of this whole thing. Now a whole bunch of districts are reliably republican, but only in the sense that A republican will win. Primary challenges actually drive these districts to be even MORE conservative then they have already been modified to be through the redistricting. So the result is a house that is much more conservative then the actual country they serve. These guys have no incentive to compromise either, because they set it up so that the only of their constituents that matter are the super conservative ones.
What that makeup of congress translates to is, the party in control decides if bills come up for a vote at all. Now, to pass bills, you have to pass two separate bills in each chamber, reconcile them into one big one in a cross-chamber committee, and then re-pass the new bill in both houses. That's a lot of work and cooperation for two groups that hate each other. So the republicans agreed when Obama was elected to not allow him to pass ANY legislation. They actually said it, you can go and look it up. They would purposefully run the country into the ground, betting that the president would be more likely to be blamed then them. This is actually a popular strategy for them, they have been shutting down the govt. as a political act for decades.
So the republicans are united only in the sense that they can agree to do nothing but cut taxes and slash spending. The democrats operate less as a party and more as a conglomeration of people elected as a rejection of insane tea party mentalities, and because their majorities are always so thin, they can't get anything done. Now you can see the party context for the actual battles they face, like...
The fiscal cliff! This actually refers to 2 different things. First is that the United States has what's called a debt limit. It's number agreed upon by congress as the maximum amount of debt the US is allowed to take out. This number is essentially meaningless, since they raise it pretty much constantly whenever they need to. Our debt is really on the ups right now - part of that is inflation, the other part is the money we spent on bailing out the economy and the wars. We'll get to what that means later. Although under Bush the republicans raised the debt limit often and without objection, last year they became allergic to it and used it as a bargaining chip to cut spending. Now, fucking with the debt limit is a bad idea, because it literally puts a big halt on almost the entire government. Technically. So the dems had to scramble to make a deal. In the end, Americans hated how it all played out on both sides, so the deal that went through everyone hated - it said if congress can't figure out how to cut spending by next year, it would automatically trigger huge, painful cuts in places both dems and repubs didn't want. So 2012's fiscal cliff was both re-upping the debt ceiling and passing something to avoid these cuts, all while still trying to save money - and in the context of the individual democratic and republican politicians and their races, which i explained a bit earlier. The result is a nearly intractable situation, since no one has much incentive to compromise.
But, they have to compromise! you say. Otherwise, the cliff! The media said! Well, the amusing thing about this so-called 'cliff' is that it wasn't. The treasury secretary has 'emergency measures' that can delay the debt limit from triggering for a few weeks/months - in fact, he's using them right now, since we never actually came to a deal on the limit. As for the triggered spending cuts (which we did avoid), they wouldn't have mattered right away anyway either. Many govt. departments, including the IRS, made statements at the end of the year saying they would NOT make any changes if a deal was not in place, as they expected one would appear eventually, if not by the deadline, and converting depts. to these new rules which would've ultimately been reversed would've been a waste of millions of dollars.
So, to recap, the parties are at serious ideological odds right now, because the dems only get elected by being broad and varied, so they can't agree on anything, and the repubs only get elected by uniting to a rigid ideology of doing nothing. However, none of this matters because the media, always looking to create excitement, scared up a big story about a fiscal 'cliff' which was really more of a soft slope.
To explain the plan B vote, which I missed the first time: Boehner was basically in this situation during the fiscal cliff. He could go in and negotiate with the president and then back with his party to find something both would agree to. The president would get the senate to support the plan, and it would pass the house through republican support. That failed. Then he tried his own plan, with the idea being they'd pass a more conservative plan on their own through the house, let it fail in the senate, and hope everyone blamed the democrats. He couldn't even get his own party to support that. So his last and worst option was to renegotiate with the president on a bill that house dems and whatever moderate republicans he could scrounge up could pass. And that's what happened, and the republicans hated him for it. He's not super good at his job, but it's mostly that, as this post is designed to explain, the tea party usually has no reason to compromise.
If anything, Obama's second term will be less partisan than it was the last 4 years, just because of the demographic shifts the republicans are facing. These tea party guys face a dilemma: stay right wing and ensure re-election until your district re-purples and you go out like the dinosaurs; or move to the middle to try and save your political career, but guarantee a vicious primary where you have to slide back to the right anyway. Only time will tell, and it's important to look at all these individual parts to understand political motivation instead of pinning it on some grand, behind-closed-doors conspiracy. Everyone's just trying to make their dollar.
Reddit user Kanpai posted this excellent summary of US politics over the past several years, explaining the "Fiscal Cliff":