Connect to another Computer.

> My other computer [...]

   The non-psychics in your audience might know nothing about any of
your computers, but especially not about your (unspecified) "My other
computer".

> [...] I used to have Lynxis [...]

   Some Linksys router or other, or something else?

> [...] and that had a little plastic piece that I had to connect to the
> other computer to pick up the signal from the main computer and router..
> [...]

   The non-psychics in your audience might know nothing about any
"little plastic piece", either.

   If that "little plastic piece" was a USB-wireless adapter, then you
should be able to use it with your Netgear router, too.  (Unless you
threw it away when got your new router.)  Most likely, it was not
something that "that [router?] had", any more than "that [router?] had"
any of your computers.

> [...] Is there some similar piece of equipment I need to hook up to
> that computer that would work with Netgear? [...]

   Keep using the same one (whatever it is)?

> [...] Sorry but I am very unfamiliar with all these things. [...]

   You might need to get more familiar with them than "little plastic
piece".  The non-psychics in your audience can't see it, and if no one
knows what it is, then progress might be difficult.