Game features to search for include Shared/Split-Screen, Shared/Split-Screen PvP, and Shared/Split-Screen Co-op.
Game tags to search for include Split Screen, Local Multiplayer, Local Co-Op, and 4 Player Local.
Ways to display a PC's screen on a large TV include — Steam Link or other wireless display technology, or just buy a long HDMI cable. While Steam Link is quick and easy to setup, I've found the lag to be quite noticeable when playing games, so I prefer to use a hard-wired video-dedicated cable.
Obviously all you need is a game that supports online multiplayer, and several laptops, and you can just all sit in the same room playing together.
Steam has a feature called Remote Play Together, which basically means that you don't have to buy an extra copy of the game for each person.
Game features to search for include Online Co-Op, Online PvP, or LAN PvP.
Game tags to search for include Multiplayer, PvP, Online Co-Op, and Co-Op.
If you have several different people who have their own Steam accounts, you can actually share games with each other using Steam Family Sharing.
I believe you need to create a new local account for each other family member you want to share with.