Let's say sniff program is installed on computer A. And let's say I am looking at a Youtube video on the computer B.
Can the sniff program see what computer B is doing?
If that's possible, witch sniffing program can do that? I use Ubuntu if that helps.
Yes, it is possible. There are several ways:
You can use tcpdump or wireshark for packet capturing.
I am assuming both hosts are located in the same LAN.
Check out Wireshark.
This can't be done on a switched network, because (by definition of Ethernet switching) on these networks each switch port only receives frames addressed to devices connected to that port; so your computer will not get a copy of the frames addressed to other devices, and any network sniffer running on it will not be able to examine those frames.
You can work around this limitation if you have some kind of control on the networking equipment and/or layout:
About the sniffing program itself, the best one around is WireShark. It's available for both Linux and Windows.
I should point out - the Wireshark solution allows you to see what packets are being sent/received. It won't let you, say, watch the same YouTube video, unless you're much more sophisticated than you're likely to want to be.
The way your question is worded it sounds like you might be more interested in whether spyware/employee monitoring can be done remotely without an agent on your machine. And that's possible, but requires a level of technical sophistication your employer isn't likely to have/want/do.