Socket is somewhat used under the hood for connectivity. For example, socket is created when you open a web page, and then socket is closed. Tor circuit is a chain of sockets, as from the transport layer. "SOCKS" is a proxying extension based on sockets. SOCKS is used by Tor to expose itself for HTTP clients, like Tor2Web, that pass initial request, it is just an interface which is very similar to socket itself.
The "generic failure" is unspecified type of error (it has no reason included), but we still can know why it happens. As Tor2Web has many clients, middle relay connection (socket) may often "close", and there are many reasons for this. A socket can also be closed by relay next to middle relay, destroying the connection while it is in use.
In all these "unclear connection chain situations" it is okay to use error message like "generic error" to specify the state: "connection is suddenly just dead" before it was finished. If you don't have specific logs from Tor daemon in Tor2Web side, no chance to find out the reason. Tor2Web reports it was unable to download the response (or finish the request):
Same may occur in surface web. It was agreed that browser should check if there was "something" received from server and just display this with no error even if connection "died", and only display error "connection interrupted" if no data yet received. Connection "deaths" often occur in 3G networks and networks with malfunctioning routers.
This could be a fingerprinting method used by bad relays or exit nodes controlled by ISPs. But no proof. Don't get paranoid much, since this situation can be interpreted as "someone reset a router when connection was not yet finished".
Tor2web is used to open hidden service (darkweb) web sites without safety/privacy, e.g. via any type of web browser, outside Tor network. When you visit sites like that, be informed, that many parties including your ISP can know where did you go. You don't need to use such services when you have Tor running on your device.