I'm a US student who's been living and studying in Quito for the past month and thought I would offer my two cents on the current situation in Ecuador. First, I would like to say that most of the US media reports of the situation here are quite overblown (the BBC has done somewhat better). Yes, there's some sort of "martial law" going on, but this does not mean that the military is stopping random civilians on the streets and infringing on everyone's rights. It simply means that the military is fulfilling the police situation because the police are, well they're the one's causing the problem.
Now, on to more of the details:
First, the origins of this. As far as I've gathered from the news and talking to various Ecuadorians, what happened was this: Yesterday the national assembly agreed on a new law governing public servants (including the police). For years Ecuador has faced issues of a notoriously corrupt police force. This new law takes away some of the special "perks" they had with their job and treats them more like other public employees. They were not too happy about this.
This morning, President Correa went to the police headquarters in hopes of working this out with words. He then was attacked by angry policemen and had a tear gas cannister explode within feet of his face (they've been replaying the video over and over again on Ecuadorian news channels). From here, my knowledge gets a little murky on the specifics. Eventually the whole police force stopped working and many of them took to the streets protesting and wreaking havoc (so far the best quote I've gotten from the US Embassy's emails here is, "Due to blockages of multiple roads and tire fires being set by police, the security situation has degraded significantly").
The main thing that has come of all of this (at least in Quito, I get the impression that things are a little more intense in Guayaquil) is exactly what one might expect with an absence of police. Thieves are more emboldened (hence all of my Ecuadorian friends have warned me against being out on the streets, especially since I'm a gringo), and most of the shops are closed (because of fear of looting). Other than this, though, there's not too much outside of the ordinary in most of the city. Near the police stations there are some little protest actions. I've seen the remains of a few tires burned in the road, though not nearly to the extent that the US news media has suggested.
As for what's to come, it sounds like the government is not going to back down on the new law. What I have been told is that they plan to strip the rioting portion of the police force of their jobs and try them as civilians inciting ruckus (which is harsher than what they might otherwise get). I was also told that the police officers who hit the president will be going to jail, which in my humble opinion sounds pretty reasonable.
The overall reaction of Ecuadorians I've talked to seems to be a somewhat lackadaisical "yeah, this sort of stuff happens. It'll blow over eventually" and a somewhat more ardent concern for my safety.
If anyone has questions (or corrections for that matter) feel free to post them in the questions and I'll do my best to answer them.