How exactly are petri dishes used (e.g. in medicine)?
I think the petri dishes you are referring to are also called agar plates. Agar is a 'growth medium' for microorganisms. It provides the nutrients that the microorganisms need to survive. It can also contain main other 'things', such as antibiotics or pH color change indicators. There are many types of agar, for example, tryptic soy agar is used to grow E. coli. I think the oven you are referring to is called an incubator. Incubators provide the atmosphere, particularly temperature, needed for the organisms to grow. There are many types of incubators. Some microorganisms actually cannot grow in oxygen rich environments, and some only grow at specific temperatures. These organisms must be incubated under their preferred conditions in order for them to grow.

And yes, a lot can be learned from simply looking at the agar with the organisms on it. What exactly can be learned depends on the growth medium (agar or other medium). 

To summarize, petri-dishes are just the plastic dishes that contain growth medium. And incubators are usually refrigerator looking things that provide the proper temperature for the organisms to grow. The goal is generally to mimic the natural environment of the organism, but as I stated, scientists often add many other things to the environment for experimental and diagnostic purposes. 

Here is a list of common agars used in bacteriology http://learn.chm.msu.edu/vibl/content/differential/