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Korean characters are like Japanese/Chinese because Korean and Japanese took their writing systems from the Chinese. Then then later created their own writing systems based on Chinese. The Koreans are extremely proud of their writing system--Hangul--created by scholars centuries ago. Some Chinese characters are used in Korean writing but it's mostly Hangul. Korean was also influenced by the Japanese annexing it and ruling over it from 1905-1945. They forced everyone to learn Japanese and punished people for using Korean. Japan acquired its writing system from China in the early first millennium. They still write using Chinese characters, but they have made them their own so while some characters are identical to the Chinese, others are a bit different. And then Japan created its own writing system to augment the Chinese--two syllabaries called hiragana and katakana.

When linguists talk about language families, they are referring to the spoken language only. Writing systems came much later. So while Vietnamese and English use the Roman alphabet, they're not considered to be in the same language family because the two languages are related only extremely distantly in human pre-history. That's why Korean, Japanese, and Chinese can have some closer ties in terms of their written languages, but that has no bearing on how closely related the languages are. Incidentally, when Japan adopted the Chinese writing system (this is around 500 AD and onwards), Chinese became the language of scholars. This meant that a lot of Chinese words crept into Japanese, and the language was forever changed as a result. Every written character has at least 2 pronunciations, one from the Japanese, and at least one from the Chinese. So Japanese has those ties to Chinese, but since they were added after the Japanese language had become developed, they don't affect Japan's status as a language isolate. This is sort of like how the Normans from France conquered England in 1066, and from then onwards, a very large amount of words of French origin were added to the English language. A lot of Latin and Greek words have been added to English in a similar fashion. But this vocabulary addition doesn't change the original language group.

Korean is completely different from Chinese--Chinese is a tonal language but Korean is not. And language structure is quite different.

Could you clarify what you mean by "characters"? If you're talking about the writing systems, it's only in the past that they were similar, and that's because they were both using Chinese characters. But that' not the case today, to the extent that it was in Japanese, or for the most part at all in Korean except for a few common abbreviations. If you can clarify what you mean there I can provide a much more detailed answer.

Until you have a chance to answer that —

There are people who try to link Modern Japanese and Modern Korean through the Altaic family, but this version of Altaic (called Macro-Altaic) isn't really taken seriously, so you can disregard any links to Altaic that people might post here in response to your question. Micro-Altaic, which is similar but doesn't include Japanese and Korean is somewhat more well received. At present, there's not widely accepted theory (at least that I've heard of) which gives them any genetic relationship. Instead, the similarities may well be the result of areal features.

Also, I'm paging /u/limetom, who can do a much better job of answering this. I'll hold off on adding anything else until he's got a chance to blow our minds.

Other people have addressed the writing system issues, so I don't really have anything more to say about that.

But about the languages.

A genetic relationship between languages is not the only reason languages can look like one another. Similarly, related languages don't necessarily have to resemble one another.

We generally find that there are four possible reasons why languages share common elements:

• Chance
• Borrowing
• Genetic relation

An chance resemblance is the situation where language A and language B share a similar element, like a sound, a word, a grammatical construction, etc., just on chance. Given the limitations of the human vocal tract, the size of words, and a number of other factors, it's actually not surprising that elements end up looking the same across languages. We can even attempt to estimate how common this might be. Ringe 1992 is an attempt at this sort of estimation. A good example of this is English mess 'mess' and Kaqchikel mes 'mess' (Campbell 2013: 111). Both mean the same thing, and both have approximately the same phonetic shape (that is, they sound more or less the same). But we have no reason to believe that this is anything other than a chance resemblance.

Borrowing is a situation where language A copies ("borrows") an element from language B. This copying can be of any sort of linguistic element--from sounds, to words, to grammatical constructions, and it doesn't have to be a perfect copy. A fairly straightforward example of this would be something like English sushi, a borrowing from Japanese sushi. A more interesting borrowing would be something like Nivkh u-, which is borrowed from Ainu u-. Both of these reciprocal markers, which prefix to a verb and mean the action of a verb happens to both the subject and the object. We'd usually translate this as 'each other' in English. For instance, Ainu u-minare 'to make each other laugh'. Borrowing can happen for a variety of reasons, but generally, when languages are in contact for a long period of time, they tend to borrow more from one another.

Finally, we have a genetic relationship. This is when two (or more) languages descend from a common ancestor. English hound 'a kind of dog with a good sense of smell, used for hunting and tracking', German Hund 'dog', Norwegian hund 'dog', and Gothic hunds 'dog' derive from a common ancestor, Proto-Germanic *hundaz 'dog'.

Japonic, the small language family which Japanese is a part of, and Korean have been in contact for a long while. In terms of pre-history, it's pretty clear from genetic and archaeological evidence that Japonic speakers entered the Japanese archipelago from the Korean Peninsula starting around 900 BCE.

Korea was, until the establishments of Japanese missions to China in the Ming Dynasty, the conduit of Chinese culture into Japan, including Buddhism, writing, and the like.

We can see a number of borrowings, especially in Old Japanese (recorded in texts up to around 800 CE). For instance, in the Man'yōshū (the earliest large collection of Japanese poetry), there are not only Korean loan words, but even a partial Korean poem (Vovin 2002). Contact continued and continues until today, though perhaps not at the same scale.

Like I mentioned above, it's likely the case that a lot of the reasons Japanese and Korean (and a number of other languages in the region) all look the same is because of borrowing of a variety of elements. For instance, many languages in Northeast and Central Asia have the restriction that words are not allowed to begin with r. In many of these languages, this gets violated, as they have borrowed r-initial words from Chinese or other languages, but in terms of their non-borrowed vocabulary, you basically cannot find words that begin with r. One example in Japanese that's often thrown around is rakko 'sea otter', but this word is suspicious. Sea otters never lived further south than Hokkaidō. In fact, this word is actually a loan from Ainu, a critically endangered language isolate spoken in the north of Japan, which, unlike many other languages of Northeast Asia, is not subject to this no initial r restriction.

The term "language area" refers to a large amount of common structure and vocabulary, as a result of borrowing, between groups of languages confined to a certain geographic area. Northeast and Central Asia is a good example of this, though perhaps a more typical one is the Balkans, where are number of related and unrelated languages all share a bunch of common features because of long-term, on-going contact.

It turns out that if we critically analyze the similarities between Korean and Japonic (for instance, Vovin 2010), it ends up being that we find borrowing--and a Northeast Asian language area, as well as chance to be better explanations.