Ethernet switch or hub or ???

> Dumb question?

   No, but describing your equipment as "my router" is less than
helpful.

> I have 2 ethernet devices and my router has one available ethernet
> port.

   What, exactly, is your "my router"?  Define "available".  How many
Ethernet ports does it have, in total?

> [...] Is what I'm looking to do even possible?

   That may depend on exactly what your "my router" is.  If your "my
router" really is a router, then you should be able to use an Ethernet
switch (or hub) to expand the number of available ports.

   If your "my router" really is a modem (only, not a router), then you
should probably add a router rather than a switch.

   Nowadays, an Ethernet hub is gettng hard to find.  An Ethernet switch
costs about the same, and performs better in most cases.

> [...] Ideally the device would come with a remote control so I could
> sit on my fat rear and do the switching from the couch.  But it's O.K.
> if I have to get up and push a switch in order to make the connection.

   No, _ideally_, the device would let you connect mutiple devices at
the same time, and not make you worry about which one is connected.  A
router, perhaps with a switch, should do that.  The only mystery is what
you already have, which would affect what you should get.