Given the constraints of this forum, I can only provide a brief answer re: the definitions of makruh:
Makruh, according to the hanafi school, is divided into makurh tahrimi, and makruh tanzihi.
Makurh tahrimi is something that the lawgiver has ordered us to refrain from through a zanni (not so categorical) evidence (i.e. khabar wahid or isolated tradition), for example: to propose marriage to a girl to whom another person already proposed; or wearing of silk-clothes by men, etc. The difference between this and haram is: the one who deniesharam becomes a kafir (infidel), while a person who denies makruhtahrimi cannot be deemed as a kafir, and the punishment for doingmakurh tahrimi is less than the punishment of doing a haraam thing.
Makruh tanzihi, on the other hand, applies to something that the law-giver has ordered us to refrain from, albeit not so firmly: an example for this is consuming the horse meat.
The legal consequence of doing makruh tanzihi is that the one who refrains from it will be rewarded; whereas one who commits it will not be punished.
The definitions vary slightly in the other schools: for instance, in theshafi‘i school: makruh tahrimi is that which the lawgiver has ordered us to refrain from it firmly but not as that of haram. The punishment for doing so is less than the punishment for doing haram; example: offering optional prayers while the sun is rising or setting.
As for makurh tanzihi, shafi’ites define it as something which the lawgiver has ordered us to refrain from it not so firmly. Therefore, a person leaving it while seeking reward from Allah will be rewarded for doing so; but he will not incur punishment if he does it. An example is: a pilgrim fasting on the day of Arafah.
For further elaborations, you are advised to take a course on usul al-fiqh under a properly qualified scholar. You may also be able to explore the issue further by reading Muhammad Hashim Kamali’s Principles of Islamic Jurisprudence.