Rimes
The pronunciation of a Chinese character can be broken into two parts: (1) the initial consonant, referred to as Seongmo (聲母, 성모) or Seong (聲, 성); and (2) the rime, which consists of the medial vowel and the final consonant, referred to as Unmo (韻母, 운모) or Un (韻, 운). All rimes can be classified into one of 106 rimes, as seen in the Riming Table below. 30 of these rimes are of the plain tone (平聲, 평성); 29 are of the rising tone (上聲, 상성); 30 are of the departing tone (去聲, 거성); and 17 are of the entering tone (入聲, 입성). These last three tones are classified collectively as the oblique tone, or Cheukseong (仄聲, 측성). The plain tone is alternatively called “low” or Jeo (低, 저); and the oblique tones are alternatively called “high” or Go (高, 고).
As an example, take the character 天, pronounced 천 (Cheon) in Korean. The ㅊ (Ch) is the initial consonant; andㅓ + ㄴ (Eon), which is of the plain tone, is the rime. In the Chinese character dictionary, or Okpyeon (玉篇, 옥편), 天 is classified as riming with the character 先, pronounced 선 (Seon) in Korean. Note that the character it rimes with can have a different initial consonant.
It should be also noted that due to Korean phonology and morphing, Korean pronunciations do not match always well with the listed rimes. For instance, the character 思, pronounced 사 (Sa) in Korean is listed as being a rime with the character 支, pronounced 지 (Ji). Furthermore, some of the rimes listed above seem redundant. For instance, 東 (동) and 冬 (동) are both pronounced “Dong” in Korean. Again, this is probably due to Korean pronunciation and morphing.
On Learning Tones and Rimes
Most, if not all, books and online resources in English on learning Hanja every mention that there are four tones. At the very least, they may mention that there are vowel length distinctions. Indeed, plain tone characters have short vowel lengths (短, 단), while oblique tone characters have long vowel lengths (長, 장). As for learning, memorizing tones and rimes is necessary to understand and fully appreciate Classical Chinese poetry, or Hanshi (漢詩, 한시). Memorizing tones and rimes is certainly helpful, but not necessary, in understanding Classical Chinese prose, or Hanmun (漢文, 한문). This is because for one character there may be different tones, even with the same pronunciation.