What exactly is a 20 amp dedicated line?

You can have it both ways; a dedicated line can be used with both outlets, or you can break the tabs connecting the upper and lower outlets and have a seperate dedicated line to each outlet, giving you 2 dedicted lines. With a dedicated line, no other outlets are on the line except the one you will be using.

What exactly is a 20 amp dedicated line?

A line that is dedicated for a specific purpose.

A 120Vac 20 amp dedicated branch circuit consists of,
(1) Equipment grounding conductor
(1) Neutral conductor
(1) Hot conductor.

A true dedicated branch circuit does not share a cable or raceway with any other branch circuit/s.

How many receptacles are connected to the branch circuit depends on what was specified by the customer or manufacture of a piece of equipment.

A single receptacle could be specified.

A duplex receptacle.

Two duplex receptacles.

One duplex is quite common.....

What did you specify when you hired the electrician?

Note.
Per 2008 NEC code if a duplex receptacle is split and fed by two branch circuits the breakers feeding the duplex receptacle shall have a common tie handle so both circuits can be de-energized by one handle.

Jea48, do you know if it's Code to run a single 220v romex line consisting of 2 hots, 1 neutral and 1 ground into a single receptical box, with each hot powering a separate receptical and the single neutral and ground doing double duty for the two recepticals? I assume that each hot would be connected to a separate breaker in the fuse box. If Code, this would be an efficient way to run 2 dedicated lines with one romex run.

Of course, the polarity of each receptical would be inverse to the other if powered by the 2 hots.

WOW! I think that I bit off more than I can chew from the responses above. Thank you very much for all of the information.