Sahih (صَحِيْح) is an Arabic word that means genuine/authentic/sound. It is used in classification of ahadith and is the highest level of authenticity given to a narration that passes all of Imam Bukhari's criteria for classification as a sahih hadith. The ahadith (sayings/narrations/reports) are contemporaneous accounts about Muhammad, his life, and the life of his companions. Whereas the Qur'an was written down in book form and later codified by Caliph Uthman, the ahadith did not benefit from such a process, but instead were passed down orally from person to person, until a collector of ahadith came to record them (often many hundreds of years later).
 Bukhari's Criteria
Muhammad al-Bukhari (Muhammad Ibn Ismail Ibn Ibrahim Ibn al-Mughirah Ibn Bardizbah al-Bukhari) was an Islamic scholar and collector of ahadith. When he set out to collect narrations he decided on a set of criterion that he would use to determine the authenticity of what he was collecting. Any narrations that he collected that did not pass these requirements would not be accepted into his collection. Because he used these criteria, his collection is known today as 'Sahih al Bukhari.' Some years later, his student Muslim ibn al-Hajjaj (Abul Husayn Muslim ibn al-Hajjaj Qushayri al-Nishapuri) would use the same criteria to guide his collection. Muslim's collection is known as 'Sahih Muslim.' These are the only two ahadith collections that are prefixed as whole by 'Sahih.' These two Sunni ahadith collections are accepted by Sunni Muslims to be Sahih as a whole, although this does not mean that every single narration included in both collections is Sahih, nor does this preclude narrations in other collections from being sahih also.
Imam Bukhari came up with 3 criteria which he used to determine whether or not a narration was sahih or not. They are:
The isnad, or sanad, is the chain of narrators. Since the narrations were passed orally from person to person, it is important to know who heard the narration from whom before themselves. So if Bukhari collects from Person D, he wants to know who person C, B & A were (of course the isnads are longer).
This refers to the reputation of each Muslim sub narrator in the isnad. Each 'link' in the chain must be established first, then the reputation/character of each 'link' must be evaluated. If they were known as a pious, honest Muslim, then the narration passes this criteria; however if one (or more) of the 'links' was a known liar, or had doubtful character, then the narration did not pass and was rejected.
The contents of the narration must not be in contradiction with the Qur'an. For example, there are narrations that state that Muhammad performed miracles; but the Qur'an makes clear that he did not - so even though some of these 'miracle' ahadith can be found in the Sahih collections, we know that they fail this criteria, so they can be discarded as 'sahih.' The exception to this rule are of course, the narrations regarded by scholars to be Qudsi. These narrations may contradict the Qur'an, but this does not mean they are not Sahih.
 Further Classification
Previous collectors of ahadith had attempted to come up with criteria to establish authority and authenticity of the different narrations they collected, but none were as stringent as Imam Bukhari's.
Mutawatir means 'corroborated.' This occurs when a narration (with minor variation or word for word) is passed down through different isnads (chains). If the same/similar narration is passed down through 20 isnads, then it is of higher authenticity than one which may be reported through only 2 isnads (especially if they contradict). This classification can be joined with other classifications; for example, if a narration were both 'sahih' and 'mutawatir' then it is of the highest authenticity regarding classification of ahadith.
Hasan means that the narration is good. It is not as good as 'sahih', but this was the standard for classification before Imam Bukhari's criteria. Of course this does not mean that the narration itself is not authentic, but that there should be supporting evidence (i.e. other stronger narrations) to support the information in it.
Da'if means that the narration is weak - usually caused by a problem or doubt about the isnad or one or more of the reporters in the isnad. Again this does not mean the narration is an outright forgery, but that it should be supported by stronger narrations.
This classification places the narration at best, as doubtful, and at worst, an outright forgery. This is the weakest of classifications.
 Example of Bukhari's Criteria
Let us use Aisha's age of consummation as an example of how a historical detail can be shown to be reliable:
The majority of the narrations in Sahih Muslim and Sahih Bukhari are narrated by Aisha herself. This places her as the 'start point' of the isnad. However let us take some creative license with who she told (isnad) and their character ('adl) so that we can better illustrate the point of Bukhari's criteria and the inherent problems with a 'telephone game' hundreds of years long.
If Aisha only passed the narration of her age onto 3 people (It's Aisha who tells us how old she is - hence "narrated Aisha"). The people and colored strips connecting them, in the diagram to the right, represent the different chain of narrators (isnad).
If there happened to be a chain that Bukhari 'caught' the end of (represented by the black strip in the diagram), that does not connect back up to Aisha, or there was a 'weak link' (an untrustworthy Muslim, represented by the red person in the diagram), then that narration would not be accepted.
So Bukhari goes to collect the narrations, and he finds 6 people who do not know each other, who report to him the exact same narration regarding Aisha - including all the details; i.e. that she was 6 & 9 when married and consummated.
 Claim of Non Authenticity
A lot of Muslims will declare any narration that they do not like as 'unauthentic.' However, since we are playing by their own rules (their classifications for ahadith), the onus is on them to show how the narrations are unauthentic. If they are dealing with Sahih Muslim or Sahih Bukhari, then they must address Bukhari's criteria and bring the evidence to show which criteria/s the narration fails on. If they cannot, then it is their duty as a ahadith accepting Muslim to accept the narration until they can show which criteria it fails.
For example, in the modern age, many Muslims find it abhorrent that Muhammad had sex with a child, so the sahih ahadith where Aisha tells you her age are often contested by Muslims and declared 'unauthentic', but let us have look at how many there are; we can count them. Bearing in mind that for each narration in each collection, it was actually passed down through different isnads. There are a lot more 'chains' than the example picture shown above. The details of the narrations are consistent across collections (ie. Tabari, Ishaq, Muslim, Bukhari, Dawud etc..) so that rules out the possibility that the collector made them up or a specific area had incorporated a cultural practice into Islam, and even one sahih narration is enough to form a part of shari'ah.
In short, with these narrations, given the plenitude of them, also the fact that they are 'Sahih' and 'Mutawatir', when dealing with ahadith, this is the closest you can get to: "yes this definitely happened." Therefore the onus is on the Muslim to use his own criteria to disqualify the narration and not on the opponent to prove it is authentic, because they are in Sahih collections.
 See Also
- Hadith - A hub page that leads to other articles related to Hadith