To protect yourself, don't download files from an untrusted network or website. Usual targets would be public WiFi or places of business where an attacker can acquire lots of information from several people quickly. Worms usually aren't a major threat for home networks. There have been quite a few major attacks by worms in the past if you want to do a quick Google search and read up on them. Worms can be spread through a number of ways: Some older worms would propagate themselves by sending copies of themselves via email, whereas others may exploit open ports on the network and send themselves using any form of transmission available. They are fully automated, meaning they don't need any input from a user to activate, and they often spread without your knowledge.
Yes, viruses can spread and infect computers simply from being on the same network.
