A: There is no doubt in my mind. The signal is exactly what you'd expect for gravitational waves from two black holes merging in general relativity. I mean this in more than one way.
First of all, the waveform is almost the same at both detectors (one in Hanford, WA and one in Livingston, LA). Actually's it's more than that. The two detectors are rotated almost 90° about the vertical relative to each other (see below). That means that the two waveforms should actually be the negatives of each other. That's exactly what's seen (from  ):
In the top right panel, you'll notice that the Hanford signal has been inverted (flipped top-to-bottom). Again this is due to the relative orientations of the two detectors:
Furthermore, the diagnostics of the detectors show that they were performing very cleanly. LIGO has literally tens of thousands of auxiliary channels: acoustic sensors, seismic sensors, radio receivers, magnetometers, monitors for the electrical grid, you name it. All of these were clean at the time of the event. None of them "vetoed" the event.
Secondly, this gravitational wave signal is exactly the signal that everyone had been expecting to see in LIGO. Well, maybe they were expecting a binary neutron star before a binary black hole, but still, everyone was expecting to see the coalescence of a binary of compact objects. This is what theorists have been modeling for years now.
I expected the first detection to be kind of marginal, just at the threshold of detectability. Nature was so kind to LIGO--the first thing it saw was a whopping big signal! You don't need particularly fancy tools to detect that it's there. In fact, with a bit of filtering, you can even just hear it out of the noise:
Technically, as a physicist, I should always have some tiny doubt in my mind. Technically, it's possible that the random noise in each detector conspired to make two waveforms with the same signal, but flipped upside down in one, within 7ms of each other. The data analysts have estimated the rate at which random noise fluctuations would make this happen: it should happen roughly once every 203,000 years (or rarer). So as a physicist, I have to admit this conspiratorial possibility. But honestly there's no doubt in my mind.
A: So much new physics! I'm sure I can't list all the questions people want to answer, but here are a few fascinating topics. Of course, these are just the known unknowns. I'm most excited about the unknown unknowns.
A smattering of topics people want to learn about from gravitational waves, more details below: black holes and their populations; properties of dense nuclear matter; gamma ray bursts; supernovae; exotica like cosmic strings; and testing general relativity. This answer turned out way longer than I expected, but there are just so many cool things to learn!
Black holes: the first gravitational wave detection (GW150914) was the merger of two black holes, each about 30 times the mass of the sun! This is what the signal looked like (from  ):
We should expect many more binary black hole (BBH) mergers to be measured in the coming years. The most important astrophysics to remember about this first detection is this: Before September, the only black holes people had ever observed were either supermassive (between 100,000 to a billion solar masses; these are found in the cores of galaxies) or between 5-15 [math]M_\odot[/math] (that's the symbol for the solar mass). There's a giant gulf in the middle! We don't know how supermassive black holes are formed.
With GW150914, we've observed a ~60[math]M_\odot[/math] black hole for the first time.
With more gravitational wave events, we will understand how often binary black hole systems are, which might answer the question of how supermassive black holes are formed.
Properties of dense nuclear matter: one of the prime targets that Advanced LIGO is waiting for is the inspiral and merger of a pair of neutron stars. Reminder: when a star is quite massive, but not massive enough to make a black hole, the remnant it leaves behind is a neutron star (NS); a NS is about 1.4[math]M_\odot[/math], and about 25km across; this is at or above the density of atomic nuclei! We don't understand nuclear matter at these high densities. There's no experiments we can do in the lab to probe the conditions in a NS, either--collider experiments that can generate these densities are too "hot" relative to the "cold" conditions of a NS (this is a very strange thing to say, since NSs are millions of degrees, but there's a technical sense in which they're cold!).
When a pair of neutron stars inspiral, they create tides on each other (just like the Moon creates tides on the Earth). Just how big these tides are depends on the properties of dense nuclear matter. Depending on how "squishy" the nuclear matter is, these tides can make the binary inspiral faster.
How quickly the neutron stars inspiral is encoded in the gravitational waveform. By observing just how fast binary neutron stars inspiral, we can learn how squishy they are and how big the tides are, and therefore learn about the properties of dense nuclear matter.
Gamma ray bursts: about once per day, one of a few dedicated space satellites will detect an intense burst of gamma rays, coming from somewhere in the distant universe. These gamma ray bursts (GRBs) actually come in two varieties, short and long. The long variety is thought to be caused by massive stars that collapse. The short variety is thought to be due to the merger of two neutron stars. However, there's no smoking gun that this is true! We don't know what causes gamma ray bursts.
If we observe a gravitational wave at the same time as a gamma ray burst, we'll have a smoking gun.
Gravitational waves could reveal the mechanism behind gamma ray bursts.
Supernovae: a supernova is understood to be the death throes of a very massive star. Fusion in a star's core creates heat and pressure, which pushes against the massive weight of the star's outer layers. Once there's no more energy to be had from fusion (once iron and nickel are made), on more pressure, and imminent collapse. However, the details between the collapse and the resulting explosion are not understood. Collapse is stuff falling inward, but this has to turn into stuff flying outward.
Looking at supernovae through telescopes isn't particularly helpful, either. The light that you get is from days to weeks to months after the explosion itself took place: here's a light curve from a supernova, notice that the horizontal axis spans 150 days!
(Fun fact: this light curve is powered by the radioactive decay of elements that are produced in the explosion, and that's what sets the characteristic timescale of the light curve).
How can we get a signal from the collapse itself?
The violent churning of matter that happens during the explosion is thought to produce gravitational waves, though not nearly as large a signal as a binary black hole or binary neutron star merger.
If we're really lucky, a supernova will go off in our own Milky Way galaxy while advanced LIGO is running, and we'll be able to see the gravitational waves from the supernova--and then in the following weeks and months, we could see the supernova with ordinary telescopes.
The details of a supernova's gravitational waves could help us understand how a collapse becomes an explosion--what powers supernovae!
Exotica: some high energy theories beyond the standard model predict something called a cosmic string. A cosmic string is a hypothetical remnant of a different phase of the universe, from close to the big bang. This model predicts that cosmic string typically make "cusps" that move at the speed of light, and emit copious amounts of gravitational waves. Currently, we have upper limits on how many cosmic strings might be around, since we've never seen any evidence for them.
Gravitational waves could providing a smoking gun for cosmic strings. Or, more likely, they will place the best constraints on the existence of cosmic strings.
There are other exotic hypotheses, like a stochastic background of gravitational waves left over from after the big bang.
Testing general relativity: this is my favorite, because it's what I spend my days trying to calculate. We want to know: Does general relativity faithfully describe our universe? Or does it need some corrections?
Binary black holes are an ideal testbed for to answer this question, because there's no messy gas/plasma/magnetic fields around that you'd need to account for--just pure, unadulterated gravity! The details of just how quickly two black holes inspiral together can encode corrections to general relativity. Those details will show up in the gravitational waveforms that advanced LIGO is measuring.
So, by very precisely measuring the gravitational waveforms, we can check if gravity is described well by general relativity, or if it needs corrections.
There are other tests, too: in general relativity, there's only one "type" of gravitational waves (with two "polarizations"). In there are corrections to GR, there could be one or more extra "types" (polarizations) of gravitational waves that aren't allowed in general relativity. With a network of three or more detectors, we could check for the presence of these additional polarizations.
Measuring the propagation speed of gravitational waves is also a key test of general relativity. In GR, gravitational waves travel exactly at the speed of light. If there are corrections to GR, the speed of gravitational waves could be different!
Unknown unknowns: before radio telescopes, we did not know about quasars or pulsars. Before gamma ray satellites, we did not know about gamma ray bursts. A similar story is true for every new type of instrument we've used to observe the universe--microwave, infrared, ultraviolet, X-ray, cosmic ray, neutrino.
I'm confident that gravitational waves will make the list, too. We've already thought of black holes binaries. We've already seen binary neutron star systems with ordinary telescopes, just not yet with gravitational waves. But we can't outsmart nature.
I hope that some day soon, somebody looks through the LIGO data and says "What's that?"
A: This is a very fancy and refined Michelson interferometer. First of all, what is a Michelson interferometer? Laser light enters at the bottom left and hits the beam splitter, half of the light going towards the top of the diagram, and half the light going to the right. After the 4km journey down each arm, the light bounces back to the beam splitter. Now each beam of light gets split by the beam splitter. The two beams that head to the left (back towards the laser) have the same phase, meaning they constructively interfere with each other, and add up. The two beams that get split heading down towards the photodetector have the opposite phase, meaning the destructively interfere with each other. So when there's no disturbance to the interferometer, there's no light landing on the photodetector.
Any light that lands on the photodetector can be read out to measure the imbalance in the arm lengths. Servo control systems then give tiny pushes to the mirrors to rebalance the system, to keep the photodetector dark. The amount of pushing that's needed to keep the photodetector dark is proportional to gravitational waves (but also to seismic noise, acoustic noise, thermal noise, and a host of other noise sources.
What happens when a gravitational wave passes by? It changes the distance between freely floating test masses in a very specific pattern. Here's how a ring of freely floating masses gets distorted as a gravitational wave goes straight through the computer screen:
But the mirrors are not "freely floating," right? Well, those test masses are suspended on a very fancy quadruple pendulum system, so that above around 10 Hz, we can think of the masses as "free".
As a gravitational wave passes by, it's going to change the separation between the test masses. That tiny change in length will change the relative phase of the light that hits the beam splitter from both arms, so no longer is there pure destructive interference heading down to the beamsplitter. A bit of light "spills out" of the arms, heading to the beamsplitter.
This is a vastly simplified explanation. In reality, each arm is a resonant cavity, and the whole Michelson interferometer sits inside the power recycling cavity. There are lots of optical cavities (the pre-mode-cleaner, the output mode cleaner, the signal recycling cavity), lots of sensors, lots of feedback servo systems to keep everything align. But the above description is at the heart of how LIGO works.