Does the El travel from Chicago Midway airport and Chicago OHare airport?

Yes.

The Blue Line travels from O'Hare to the Loop (downtown) in about 45 minutes for $2.25 (as of 2009) per person.

The Orange Line travels from Midway to the Loop in about 30 minutes for the same price.

There are no lines that go directly between Midway and O'Hare, however you can transfer between the Orange and Blue Lines at the Clark/Lake station for free.