If the ADSL modem router provides multiple wireless SSIDs how does this offer security? Is it possible to have multiple subnets with SSIDs?
Usually the multiple SSID option on a router leads to multiple VLANs, or some other method of separating the traffic between the two networks provided by two SSIDs. Often this is used to provide a guest network and a non-guest network, and providing some assurance that the guests cannot access the non-guest portion of the network.
Sometimes multiple SSIDs are simply there because the router has multple radios (say one at 5Ghz and one at 2.4Ghz), these are not normally separated and will not be secured from each other.
The purpose of multiple SSIDs is to create distinct wireless networks that can have different security properties. For example, you can create an SSID "HOME" that allows file access and printer sharing and another SSID "GUEST" that only provides Internet access. This keeps your guests from getting access to your machines.
It also allows you to change the encryption key for the "GUEST" network as often as you want (if you gave it to someone you no longer fully trust or just wish to rotate it regularly) without having to change the keys in all your permanent, trusted machines.
SSID don't offer security, encryption and authentication do. I'd use the term vlan very carefully for wireless, it should not be confused with a layer 2 wired vlan. Install kismet on your old laptop, and see what you can see.