
  This is an attempt at a canonical question following this discussion on Meta. The aim is to produce basic answers that can be understood by the general audience.


Let's say I browse the web and use different apps while connected to the network at work. Can my employer (who controls the network) see what websites I visit, what emails I send, my IM messages, what Spotify songs I listen to, etc? What are they able to see?

Does it matter if I use my own computer, or one provided for me by my employer? Does it matter what programs I use, or what websites I visit? 

Yes. Always assume yes.
Even if you are not sure, always assume yes. Even if you are sure, they might have a contract with the ISP, a rogue admin who installed a packetlogger, a video camera that catches your screen... yes.
Everything you do at the workplace is visible to everyone. Especially everything you do on digital media. Especially personal things. Especially things you would not want them to see.
One of the basic rules of Information Security is that whoever has physical access to the machine, has the machine. Your employer has physical access to everything: the machine, the network, the infrastructure. He can add and change policies, install certificates, play man in the middle. Even websites with 'SSL' can be intercepted. There are plenty of valid reasons for this, mostly related to their own network security (antivirus, logging, prohibiting access to certain sites or functionalities).
Even if you get lucky and they cannot see the contents of your messages, they might still be able to see a lot of other things: how many connections you made, to which sites, how much data you sent, at what times... even when using your own device, even using a secure connection, network logs can be pretty revealing.
Please, when you are at work, or using a work computer, or even using your own computer on the company network: always assume everything you do can be seen by your employer.
