I hate to nitpick, but I'm afraid that the account I am currently reading differs in some key aspects from Segnbora-t's account of the war. Rather than rebut Ms. t's account of things point by point, I believe I'll just throw in my own account of the war down here, since that's less tedious for me and involves less scrolling up and down the page by you, the reader.
First of all, key to understanding this war is the fact that China had traditionally been a hegemonic power in its section of the world, enjoying the subjugation of a number of tributary states, including both Japan and Korea at various points, as well as Vietnam. In practice, the relationships don't seem to have been far different from the way that the Roman empire had conducted its foreign affairs during its early days or the way the U.S.S.R. conducted its in later years--i.e. one central state, surrounded by a number of nominally sovereign states that just happened to depend on the central state for most things. Of course, the Chinese emperors tended to overinflate their own importance and often claimed that they were the sole source of civilization in the entire world and that no one else could possibly be their equal. This provided imperial legitimation, and consequently threatened that legitimation when those ideas were threatened.
While China was involved with the Sino-French war and two concurrent Islamic rebellions to the northwest, some bad things transpired in Korea. In 1875, a Japanese survey crew who just happened to be heavily armed had a shootout with some Korean troops. After this conflict, the Chinese office of foreign affairs, the Zongli Yamen, said that they weren't responsible for anything Korea did. This was a little surprising considering their traditional stance toward Korea, but none too surprising at all considering the state of the beleaguered Chinese army. Japan then concluded a treaty with Korea as though Korea was a sovereign nation-state like any other. China, afraid Japan would just take Korea (which is a great place from which to invade China, geographically), then tried to reassert influence by establishing a Chinese Resident in Seoul and encouraging Korea to open up relations with the nations of the west, who were in favor of these moves. Korea was thus exposed to the wide world and in a very dangerous position all of a sudden.
The Koreans didn't know what to do about this exactly, but it wasn't long until two movements came together around the queen and the former regent, known as the Taewongun, respectively. As is natural when two movements are formed in a society, these two were diametrically opposed on the issue at hand, namely whether to seek help from Japan or not. The queen said yes, while the Taewongun disagreed forcefully enough to stage a coup against her. Although the coup failed, seven Japanese officers were killed. Afterwards, to prevent the Japanese from punitively occupying Korea, China took and held the Taewongun in Chinese territory. All the same, they were in no position to say anything to the contrary when Japan decided to station a whole load of troops in Seoul to protect their delegation to the city. Internal struggles went on in Korea, and more troops, under the command of Yuan Shikai who would later serve as president of China, were dispatched to Korea to supervise government. Japan got all buddy-buddy for a little while, and the two countries briefly cooperated to suppress unrest in Korea. That all ended when the Russians got into the act, though.
Said Russians siezed Port Lazarev, which in turn ensured that the British, who were very wary of expansionist Russia during this period, would sieze Port Hamilton. The Japanese got a little freaked out at this point, and acted like they were just going to go home. They even began to tell China that it should take Korea completely. However, in 1894 the Japanese turned a rebellion into a key opportunity to seriously fuck China over.
The Tonghak, an anti-Christian sect in Korea, led a rebellion. The Korean government asked China for some help, and Japan encouraged Yuan Shikai to attack the rebels. Japan, however, duplicitously sent in troops to reinforce the Tonghak before Yuan Shikai could get there. Japan had wanted to provoke a war, and they got their wish: war was pretty much under way. No one says much about the Chinese performance in this war because it was virtually a non-issue. They got their collective ass handed to them both on land and sea, and I'll give you five dollars if you can make a convincing case otherwise.
The Japanese public expected that they would recieve concessions in proportion to how bad they beat the Chinese, but they were wrong. Although their government had designs upon Korea, Manchuria, Taiwan, Shandong, Jiangsu, Fujian, and Guangdong, they wound up settling for asking for Liaodong and Taiwan only. However, the Russians, French, and Germans all told Japan in no uncertain terms to drop Liaodong before they got hurt. The Japanese complied. In the end, the Japanese wound up with only Taiwan and the Liuqiu Islands off the northeast coast of same, as well as the right of Japanese nationals to manufacture things in China, the opening of the ports of Chongqing, Suzhou, Hangzhou, and Shaxi to trade, and an indemnity of 230 million taels to be paid to Japan. Although I'm not sure exactly how much money 230 million taels is, suffice it to say that each tael was a big hunk of silver on which the Chinese copper currency was based. To lose 230 million of those did not feel good to the Chinese economy. Oh, and Korea became independent, too.
Up to this point, China had adopted a policy of limited modernization via incorporation of small amounts of Western technology and equipment. This crushing defeat demonstrated to the Chinese public the fact that this would not work, and helped pave the way for the reforms of 1898. To the rest of the major powers, it appeared that China was going to fall apart the way the Ottoman Empire had slowly been doing for most of the nineteenth century, so each one decided to stake out a sphere of influence in China so they would be able to keep up with the Joneses if China went under completely.
China really, really lost this war. They couldn't have done much worse if their entire army had stayed home playing Super Nintendo, although how they would do that is a mystery to me. However, this major defeat served as a wakeup call to the Chinese. They were made to realize that isolationism wasn't going to see them much further into the future and that they were going to have to adapt to the new world in which they were living if they wanted to survive as a sovereign nation. A lot of options would be assessed during the next couple of decades as a result, and for better or for worse the Communists would eventually provide the most lasting answer to the questions this war raised.
All facts and most interpretations in this writeup were taken from Jack Gray's Rebellions and Revolutions: China from the 1800s to the 1980s