Car shut off at red light, all electronics also died.

Today, when I went to start my car it would not even crank. I jumped it once,
and it started but then shut off again when I shut the hood. The second time I
jumped it, I tried driving it to a local mechanic, but it shut off at a red light.
Even the emergency lights would not come on. I drove it over to a safe spot,
and the lights came back on, but then shut off entirely. There was a lot of
corrosion on the battery. Any idea what the problem might be?

Also, when I would take it on long drives, if I stopped the car for ten minutes,
it would be hard to get it to start again, the rpms would dip low and then back
up.