These are pretty good books.

1.  The Army of the Potomac trilogy, by Bruce Catton.  Mr. Lincoln's Army, Glory Road and A Stillness at Appomattox. I first learned about the Civil War in these three great books by Bruce Catton.  That was 30 years ago.  I've gone back and read them three times since.  They lose nothing in the re-reading.  The focus is on the Army of the Potomac, it's soldiers, and it's battles.  There are images in those books that Catton created for me of the battles that I still see in my minds eye.

2. The Civil War, by Shelby Foote.  Ready for over 3,000 pages of Civil War?  If you are, this is for you.  Foote the writer is so good that it never gets tedious.  He tells the story of the whole war.  It's the most thorough work on the war you're likely to read.  Actually, there are bigger works; Time Life has a series of books, The Official Records, both are excellent, but nowhere near as readable and enjoyable as Foote's books.  The focus is military, but the political leaders are here too and it turns out that Mr. Foote has a soft spot in his heart for Jefferson Davis.

3.  Battle Cry of Freedom, by James McPherson.  This is the best one volume history of the war.  It's not just military history either.  McPherson does politics as well and he does both very well.  It's a big book, over 800 pages, but you won't notice.  It's great historical writing.

4.  The Approaching Fury and The Whirlwind of War, by Stephen Oates.  Ready for something different from the usual narrative history?  Try these two books.  Mr Oates tells the story of the coming of the war, and of the war years in the voices of the participants.  First person voices as he takes on each personality.  Lincoln, Davis, John Brown, Grant, Lee, Booth and others.  It took a little getting used to at first, but soon I realized I was enjoying it totally.  The approach removes the storyteller and lets you hear from the participants directly.

5.The Centennial Civil War trilogy, by Bruce Catton.  The Coming Fury, Never Call Retreat and Terrible Swift Sword. Three more from Bruce Catton tell the story of the whole war.  The first book, The Coming Fury focuses on the political workings of 1860, leading to the attack of Fort Sumter  and is the best book on that topic I've read.  The next two are excellent histories of the war years.  Not the scope and depth of Shelby Foote or as personal as his own Army of the Potomac series, but very good.

Also:

Ashes of Glory, by Ernest B. Ferguson....Civil War Richmond.
Reveille in Washington, by Margaret Leech....Civil War Washington, D.C.
None Died in Vain, by Robert Leckie....A good overview of those years.
The Compact History of the Civil War, by Ernest and Trevor Dupuy....The key is compact, the whole deal in 460 pages, and quite good.
The Civil War, an Illustrated History, by Geoffrey Ward, Ken Burns and Ric Burns....  This is the book that went with the excellent PBS series in 1990.  See the videos if possible, they are great.  The book, it turns out is nicely readable an worth the price for the photos alone.
The Civil War in Depth, by Bob Zeller....This is for the photography lovers.  The book shows the photography of that era as it was seen then, in 3-D.  The plates were stereoscopic.  Included with the book is a viewer.  I loved it.  I'd seen all the photos before, but not like that.