That's a CTB (Conditional Turn Battle) bar, like FFX, Xenosaga, or Phantom Brave. It tells you the turn order. Different actions may change the turn order - spells may make it come later, while defending can make it come sooner. It allows you to use some strategy in battle.
Yes, I did sprite the rats.
EDIT: Post while I was typing. :p
Battles are on maps, like Chrono Trigger. When you trigger a battle, the party members (up to three) will move to predetermined positions on the map. The CTB bar works as I already outlined. You can attack, defend (and restore HP), use a skill, or use an item for a turn, just like a standard RPG.
There is no escape once battle is initiated, but monsters won't regenerate unless you enter a town and leave and they're visible encounters, so you shouldn't have a problem staying alive.
The thing over the girl's head is a ring menu, it both indicates whose turn it is (if her picture on top isn't enough) and gives you the four main options.