I finally got it through my head today the difference between adding a port to a VLAN in tagged vs. untagged mode.
Tagged means that traffic with the VLAN ID already attached—that is tagged traffic—will be passed. Untagged means that traffic without the VLAN ID already attached will have the VLAN ID inserted—that is, it will be tagged—and then passed. That way if there is a device that you want to be on a particular VLAN, put the port the device is connected to into untagged mode for whatever VLAN you want, and the traffic from that device will be tagged with the VLAN ID. Tagged ports are trunked ports, that is you can assign multiple VLANs to a port and it will allow all traffic with the appropriate tags to pass.
Please feel free to correct me in the comments if you think this is incorrect.