How are transactions broadcast to the rest of the network? If a person sends a transaction to another person, and Bitcoin is a peer-to-peer application, how will the transaction get to the rest of the Bitcoin community? How does everyone have the same blockchain?

Does every system know about every other system in the network? If so, how?
If a person sends a transaction to another person, and Bitcoin is a peer-to-peer application, how will the transaction get to the rest of the Bitcoin community? 


Think of it like how gossip would spread. There are a bunch of people (nodes), and when one of them knows something, they tell it to the few (8 or so) people they are near (connected with), and then those people tell a few more people the message that was told to them, and then... Eventually everyone knows about the original information (transaction). 

So when we say bitcoin is peer-to-peer, we mean it's just nodes that have internet connections (on port 8333 usually) with other peers. Not even necessarily the one you're sending the coins to!


  How does everyone have the same blockchain?


After peers exchange their information (and validate that all transactions and blocks follow the protocol rules), they store it. Since all of the bitcoin transactions from the beginning of the bitcoin block chain are needed to validate everything that has taken place, essentially all full nodes store the same data (about 27.5 GB of pure block chain data as of now). 


  Does every system know about every other system in the network? If so, how?


No need to know about every other system on the network, each node just knows about the 8 or so nodes that it is connected to!