As much as I feel weird about recommending a book from the 1960s that only uses U.S. and Western primary sources, I think the US Army Historical Series on Korea is really good (and is free online). http://www.history.army.mil/html/bookshelves/collect/usakw.html

South to the Naktong, North to the Yalu is particularly well written, although all the U.S. Army books are slightly dense reading because they are written for an audience of army officers.

I disliked Halberstam's book, "The Coldest Winter" because it was too focused on MacArthurian melodrama and skimpy on the actual conduct of the war and had disappointingly few sources regarding the NK and Chinese side of the story for a book written in the last decade.

The book that I found readable with all the important highlights was Korea: The Limited War by David Rees. He has lots of good maps, hits all the high points while also offering some historical background for the major players (MacArthur, Eisenhower, Truman, etc.) but it's age does show (written in the '60s) and while I'm not sure how many primary source documents are available from the Northern side, in the '60s, there were certainly less, so it's written almost entirely from the American/UN perspective.

Korea: Canada's Forgotten War by John Melady, on the other hand, is a very intimate look at the war from the ground. It's a kind of anthology of a very specific point of view in the war (from the Canadian perspective, obviously). A lot of the chapters aren't really related to each other, hence why it's an anthology in my mind. The last chapter is even just a collection of wartime vignettes that aren't big or long enough to make into a full chapter. I don't say that to diminish it: some of the stories that stick out in my mind from this book come from the smallest sections.

A book I haven't read yet, but is definitely on my list is The Coldest War: America and the Korean War by David Hallberstam. From what I can tell, it's even more detailed and up to date than Rees.

There's three other books I'd suggest if you're looking into "the context of the current situation." My favorite book on modern South Korean society is Korea: The Impossible Country by Daniel Tudor. I'm not sure what to qualify this book as (a sociological survey of modern Korea? A cultural introduction?) but it's the book that gave me the best perspective on what my neighbors and coworkers were thinking when I lived in Korea. (Tudor himself was a business writer living in Seoul for years, hence his perspective comes from that. The war, the possibility of Reunification and all that that means, certainly weighs heavily on the Korean psyche, and he analyzes that.)

The two books that gave me fresh perspective on North Korean society and North Koreans were Nothing to Envy by Barbara Demick and Target North Korea by Gavan McCormack. The former picks up in the early '90s and covers the apocalyptic famine that gripped the country, while the latter covers the strategic decisions North Korea took in the '90s and early '00s. Nothing is a classic on North Korean studies, though Target is written as something of a timely political analysis, and by the time I read it in 2015, it was already out of date (but a very interesting walk through the perspective of a decision-maker in Pyeongyang).

One I would recommend is "China's Road to the Korean War: The Making of the Sino-American Confrontation" by Chen Jian (1994). It is well researched based on previously confidential Chinese archives and provides a rare perspective from the other side of the war. While it doesn't delve too much into the actual warfare, it provides an excellent overview of the background and politics that led to the war, as well as the complicated relationship between North Korea, China and the USSR (in short, Kim Il-Sung, Stalin and Mao were all staunch nationalists, and each acted in the best interest of their own countries, which more often than not pissed off the others, leading to a very uneasy alliance).

I recommend Allan R. Millett's Korea trilogy (only 2 of 3 books published so far) The War for Korea, 1945-1950: A House Burning The War for Korea, 1950-1951: They Came from the North

The 2nd book is pretty lengthy (over 600 pages), and you may want to read these after you get a general history as it moves from decisions on the battlefields to field headquarters to national capitals - from my understanding though (I haven't read a lot of other books on the subject) this one delves into the South Korean experience moreso than other books.

"The United States Air Force in Korea" by Robert F Futrell is a blow-by-blow account of the USAF and its efforts in Korea on a near day-to-day basis. It's relatively light on analysis but high on detail, and provides a picture of Air Force thinking in the immediate aftermath of the Korean War.

To the last round by Andrew Salmon is a great read. It tells the story of the Battle of Imjin- a heroic stand by a British Brigade against an entire Chinese army, breaking the back of their advance. I cant recommend it enough.

The Korean War by Max Hastings is also pretty good. Very well researched and easy to read.