I have a work computer that I sometimes bring home so that I can get work done. I know that I can't use that computer for personal reasons. My question though is, if I'm using my own personal computer can my work computer pick up what sites I've visited because it's on the same network? Then, will it still keep that information on it when I bring it back to work, even though it's no longer connected to my network?
While the answer below is generally correct, keep in mind that some websites track your browsing activity using sessions, and this data will be saved even if you change computers... The reason I think this could be important to you is that things like your search history using Google may be transferred to your work account if you use one account on both computers... Under normal circumstances, it should not be possible to view this data across the network - but it can be seen physically at your work place.
edit: the answer above is not precisely correct, the truth is - your data across the network is not separate at all, it's possible to see any data sent across the network even from a separate computer (albeit, some of it may be encrypted, ssl, etc...). Under reasonable circumstances, though, this should not happen in your case.
The following adapts only if you use some WIFI technology like WPA2. I guess that this is basically the network you are talking of.
First of all, generally your work computer will not be able to see the network traffic of your personal computer (assumed WPA2 or similar encryption is in place at the access point).
Nevertheless from a cryptographic/protocol perspective a malicious device could do so in many cases. If you use WPA2 "Personal Mode", like most private access points do, there is a shared secret that authenticates every device in the network. I assume, you use WPA2 Personal and your work computer was logged into the network with the shared password for all your devices.
WPA2 in Personal Mode does a specific Handshake when you login to the network in which your computer exchanges a unique pairwise transient key (session key) with the access point using the Pre Shared Secret (basically your WIFI-Password and some randomness). If another device did not record this handshake, it won't know the exchanged information and cannot decrypt your network traffic later on.
If the above handshake is recorded by a device which, in your case, already knows the Pre Shared Key (derived from your password), it can compute the same key as your personal computer and decrypt all traffic on layer 2 (HTTPS remain encrypted, but DNS requests and URLs etc. might not be).
Nevertheless, this would mean that your work computer has "malware" andor similar sniffing tools installed on it and I do not see any reason for doing so from your company's side.
No. The network traffic from your personal computer is not shared with your work laptop, so your browsing activity on your personal computer is entirely separate.