Conditional expectation with respect to a $\sigma$-algebra

Could someone explain what it is that we are intuitively trying to achieve with the definition? Having read the definition I could do the problems in the section of my book, but I still have no intuitive idea of what the definition is trying to achieve. When given just a single event, I understand that the definition should be the integral in terms of the restricted measure on the event set,

$$\mu_E(A) := \mu(A\cap E)/\mu(E).$$

It's also intuitively clear what information a single event carries, i.e. "the outcome was one of these in the set event set".

Can someone explain to me the following:

1. What do we even mean by the information carried by a $\sigma$-algebra? In other words, I can't even understand what we would like this to represent.

2. Why do we want the conditional expectation to be a random variable? I assume this might follow naturally from (1) if I understood what we're trying to accomplish.