What was the order of (who were the and in what order) the COMMANDING
Union generals in the Civil War (appointed by Lincoln).  Why was each
removed from command and replaced with another (what mistakes did they
make and in what battles)?  What kind of general was Lincoln looking
for.

Are you asking for a term paper outline? Mr. Lincoln had a very bad
time finding a general who would actually fight. With McClellan, he
had a general officer who know how to build an army but not how to use
it (and a disloyal general officer at that). With Burnside, he had a
general officer who could snatch defeat from the jaws of victory. Any
good history of the Civil War would give you this information.

You really have not read up on the Civil War. My ancestor died at the
Battle of Mill Springs, Sunday 19 January 1862 (Confederate side)
alongside the alphabetically last (and probably least} of all generals
of either side, Felix Kirk Zollicoffer. Lt. Gen. the Prince Polignac
is a fun, obscure general too.

No, I am not looking for a term paper outline.  However, if you would
be willing to provide that information--which, incidentally, I have
NOT been able to find in books nor online--I would greatly appreciate
it (the information in the original question I asked)

Hi Brian,

This looks like it could be a start:

http://sunsite.utk.edu/civil-war/generals.html

And this too:

http://www.teacheroz.com/Civil_War_Battles.htm

or

http://americanhistory.about.com/od/civilwarpeople/

and many others found with  "civil war generals"

IMHO, Waukon has suggested the basic problem.  The Confederates knew
they were inciting a war and that they had to defend their homeland, a
greater motivation than the Union forces and officers had, who were
"only" fighting to defend an entirely new principle about states not
being able to opt out of the Union.
Slavery was an issue too, of course, but defending it as necessity of
one's own economy is again a greater incentive than fighting to
eliminate it as a moral principle  (really only after 1863 with
Lincoln's proclamation).

And then there is the factor that the whole generation of commanders
had blessed little experience, only gaining it as the war progressed. 
U.S. Grant might have failed if he had been promoted earlier.

"Weyell, that's just ma opionion.  Ah grew up in da Deep South where
veterans of WW I were still addressed as 'Major ...' or 'Colonel
...'."

Myoarin

McDowell ? Defeated at First Bull Run aka First Manassas
McClellan ? Accomplished nothing in the Peninsula and Seven Days
Campaigns, withdrew back to Washington.
Pope ? Defeated at Second Bull Run aka Second Manassas
McClellan ? After victory at Antietam aka Sharpsburg McClellan did not follow up
Burnside ? Defeated at Fredericksburg
Hooker - Defeated at Chancellorsville
Meade ?  Victory at Gettysburg, not demoted but Grant placed above him