What does it actually mean for an electron to be excited? I've been confused on the connection between photons and electrons for a very long time.
Some examples of questions I"ve asked are linked here:
What produces higher frequency light?
Frequency of light versus frequency of electron vibration
In their answers, when people address photons, they always talk about the electrons jumping to higher energy states and dropping to lower energy states.
But...what does this mean? Does a higher energy state mean that the electron is vibrating faster? That it's further away from the nucleus?
What ARE energy states, and why do changes in them produce photons?
What I've gotten from my other questions is that electron vibrations on one atom cause other electrons elsewhere to vibrate, and the transmission of that vibration from one electron to another IS a photon...so that means a change in an energy state the same thing as an electron vibration?
 A: 
What ARE energy states



*

*First, think of an electron as a wave. We are at a scale where particle-wave-duality matters. The electron has a periodic wavelength.

*Then think of the electron as a particle. It orbits the nucleus periodically.
These two periodic behaviours must match. Let's clear this out:
I personally like to think of an electron as both wave-and-particle by imagining that it "moves so fast" that it "smears out as a stretched probability cloud" all around the nucleus. As if it is a "cloud" that "reaches" all around the nucleus.
Now, if it "reaches all the way around" and "meets its own tail", then it must end exactly as it started. Its "position" in its wave behaviour must be the same to start with as it is after exactly one full round (and again as it is after two rounds, and three and...). In other words: The orbital period must be an integer-multiple of the wavelength.
If this is not the case, then you would see an unstable electron. It would wobble around turbulently, changing its orbit randomly, until it finds a stable orbit that matches by being an integer-multiple of its wavelength. Thus, all electrons will have found a stable orbit.
The "orbital speed" needed to match such integer-multiple orbit requires the electron to carry a certain amount of energy. It cannot be in-between two integer-multiples of an orbit, because that would mean an unstable orbit, so it will have to absorb (or release) exactly enough energy to "jump" the entire way to a new motion and thus a new type of orbit.

why do changes in them produce photons?

Such an absorbed/released energy "chunk" is called an energy quantum. We have now established (with abstract analogies) the core idea of quantum mechanics. An energy quantum that is released will be sent off and leave the atom. We call this a photon. A photon is thus just an energy "chunk" or energy "package" on its way to a new home.
The stable orbits that the electrons find are called energy states. There are a few different variations that influence energy states; one is the "jumping distance" between orbits that we have described, another is the electron spins, since electrons with different spins are said to be at different energy states. There are in total 4 different so-called quantum numbers that determine the energy state.
This was the essence. 
A: 
...when people address photons, they always talk about the electrons jumping to higher energy states and dropping to lower energy states. But...what does this mean?

Imagine electrons in atoms as some volumes attached to the nucleus. Absorbing a photon, the volume is moving away a little from the nucleus. Tighten to the nucleus the electrons charge is somehow neutralized by the charges of the nucleus. Absorbing photons, the electron moves away from the nucleus and its charge emanates more and more. At the end the electron leaves the nucleus.
To extract the next electron needs more energy because the nucleus now has more protons as the remaining electrons and tighten stronger to the nucleus.

Does a higher energy state mean that the electron is vibrating faster?

If you like the image that something vibrates, then what vibrates are the atoms from the photon exchange. At any temperature above 0 Kelvin, the atoms continuously emit and absorb photons. Photons have a moment and for each absorption and emission the atom is moved. Only at 0 Kelvin do the magnetic properties of the atoms outweigh the oscillations and a Bose-Einstein condensate can form.

What ARE energy states, and why do changes in them produce photons?

Energy states are a synonym for different distances of the electron from the nucleus.
The absorption of photons lead to well defined distance changes from the nucleus, these distances are discrete numbers. Photons of some similar energy content move the electron to the same energy state (the surplus energy gets re-emitted). Photons of higher energies allows the electron to go to a higher distance from nucleus.
The better observable phenomenon is the inverse process. During cooldown of atoms, the approach of the electrons to the nucleus happens with the emission of photons with defined energy content. The emission spectrum has well defined peaks.
