All perfect praise be to Allah
Firstly, i should admit, i am not even a student of hadees, but still i will share what i know. Feel free to correct me where i am wrong.
As in your question you have specifically asked according to hadith science, so i will stick with it.
There are four major procedures to resolve contradiction (تعارض) ta'aaruz(we will use this word from now on), which are:
Tatbeeq-تطبیق: (Getting the meaning which makes both of these ahadees compatible, harmony in meaning is researched between such ahadith.)
Nasikh/Mansookh-ناسخ/منسوخ: (Determining which hadith is from the time before the second one, and consider the new one as replacement of old one) This has its own complexities, types and explanations. of-course, all of this is so complex that's why we call it hadees science.
Tarjeeh-ترجیح: (outweighing on basis of the standards of hadith and narrations) read about tarjeeh in detail here. If in a case it could not be determined which hadees is more strong or preferable and both seems equivalent in their weights, than Tawaqquf is used. Read details in the link i provided above, it has very long explanation. Read that link before asking anything about this, because this rule has many complexities and exceptions etc.
Let me quote a beautiful example for this concept of tarjeeh:
A good example of this is the timing of ‘Asr Salah. According to the majority of scholars including Imam Malik, Shafi’ee, Ibn Hazm, Imam Ahmad and many others including the students of Abu Hanifa, ‘Asr time starts when your shadow is equal to your length which means it starts at an earlier time but the opinion of Imam Abu Hanifa is that it starts later when your shadow is double your length.
Now when you look in the evidences, the opinion of the majority seems stronger so one should follow that. This is called tarjeeh where we give preference to one opinion over others.
Now from the Hanafi perspective they will look for the opinion of Abu Hanifa and compare with Abu Yusuf and Imam Muhammad. This is referred to as tarjeeh within the math’hab.
Although above example is for tarjeeh in general, but same concept apply in tarjeeh in hadees science. We follow the one which has some sort of superiority over the other.
Twaqquf-توقف: If all of the above seniors fail to resolve the issue, than we use tawaqquf, which means we will not take either of these conflicting meanings, and we will wait, until there comes something which resolves this ta'aaruz in these ahadees. Than we will follow one according to above mentioned methods.
PS: There is a difference in which order these techniques will be used among different schools of thought, for example ahnaaf have this order(as of my knowledge): naskh, tarjeeh, tatbeeq/taufeeq, tasaqut-تساقط(don't follow any of them). While ahlul-hadees or salafi school has this order(generally): tatbeeq, naskh, tarjeeh, twaqquf.
This is a different debate about their order, but the point is that these are main procedures which are followed in this case when there seems to be a contradiction or ta'aaruz.
ٰ(I will try to add examples and more links with time(when i have more time), and edits or suggestions are always welcomed. For now, this is enough i think.)
Allah knows best
