Imaam at-Tirmidhi رحمه الله often uses the term hasan saheeh to describe a hadeeth. This term has led to some confusion among the scholars. However, before getting to that term, one must first define the terms saheeh and hasan.
In general, for a hadeeth to be called saheeh it must meet the following five conditions:
(i) The chain or isnaad must be connected, with no missing links.
(ii) Each of the narrators must be a person of religion and integrity.
(iii) Each of the narrators must be proficient in narrating hadeeth.
(iv) The narration cannot contradict stronger authorities or narrations.
(v) There must not be any hidden damaging defect or illah in the transmission of the hadeeth.
For a hadeeth to be hasan on its own merit, it must also meet these five conditions. However, the level of proficiency (condition iii) is lowered somewhat for the case of hasan ahaadeeth. That is, if a person is at the highest level of proficiency, his ahaadeeth will be called saheeh. If he is at the next level of proficiency, his ahaadeeth will be called hasan.
Hence, a hadeeth could be either hasan or saheeh. What then is the meaning of Imaam at-Tirmidhi's رحمه الله phrase hasan saheeh? This phrase seems to join together two mutually exclusive terms. Unfortunately, it seems, Imaam at-Tirmidhi رحمه الله himself never actually defined the term. Hence, the scholars have tried to determine the meaning of it. They have come up with seven different ideas as to the correct meaning of Imaam at-Tirmidhi's رحمه الله expression. Here are the different explanations:
(1) Imaam An-Nawawi رحمه الله, in his Taqreeb, said that hasan saheeh is a hadeeth that is related through one chain which is saheeh and through another chain that is hasan. However, Imaam Ibn Daqeeq al-'Eid رحمه الله said that this is wrong as Imaam at-Tirmidhi رحمه الله sometimes writes, "hasan sahih ghareeb, we do not know it through any other chain."
Some later scholars have defended Imaam an-Nawawi's رحمه الله definition and stated that ghareeb, in Imaam Ibn Daqeeq al-Eid's رحمه الله quote from Imaaml at-Tirmidhi رحمه الله, refers to the fact that one of the narrators is solitary and not that there is only one chain for the hadeeth . This clarification leaves Imaam an-Nawawi's رحمه الله definition as a possible explanation except if this definition is what Imaam at-Tirmidhi رحمه الله meant he should have said "hasan and saheeh" instead of "hasan saheeh." 
(2) Some say that the hadeeth is hasan (which lexically means, "good"; here the implication is that it gives glad tidings and so forth) with respect to its text and saheeh with respect to its chain. Others argue that this does not make any sense as Imaam at-Tirmidhi رحمه الله sometimes used this term for ahaadeeth concerning the hell-fire, law of retaliation, punishment and so on. In such cases, what does it mean to say that the hadeeth is "good"? Furthermore, all of the words of the Prophet صلى الله عليه وسلم are good.
(3) Imaam Ibn as-Salaah رحمه الله has a similar interpretation. According to him, Imaam at-Tirmidhi رحمه الله used the term hasan according to its common lexical meaning while he used the term saheeh according to the hadeeth scholars' special terminology. In the same way, the argument continues, Imaam Ibn 'Abdul Barr رحمه الله once said, "This hadeeth is very good [hasan]," about a hadeeth that had liars in its chain. Obviously he was not describing the degree of the hadeeth but was only discussing the meaning of the hadeeth. 
(4) Shaykh Ahmad Shaakir رحمه الله offers another possible definition. He theorizes that hasan means accepted, that is, one should act in accordance with this hadeeth, in the same way that Imaam Maalik رحمه الله used to use the phrase, "Our people act in accordance with this report." If Imaam at-Tirmidhi only uses the term saheeh then it is similar to Imaam Maalik رحمه الله saying, "We do not act according to this report," that is, the report may have been abrogated or it does not describe a general ruling. 
(5) Imaam Ibn Katheer رحمه الله says the term refers to another new category of hadeeth between saheeh and hasan. It is the highest degree of hasan and the lowest degree of saheeh. Concerning this statement, Imaam al-'Iraaqi رحمه الله says that there is no proof for it and it is, in fact, very far away from the correct meaning. 
(6) Imaam Ibn Daqeeq al-'Eid رحمه الله also presents his own definition. According to him, the term hasan saheeh refers to a hadeeth that can be raised to the degree of saheeh as it has the best quality narrators for a hasan hadeeth. At one place Imaam ibn Hajar رحمه الله wrote something similar to this when he stated that it refers to a hadeeth which has some narrators of hasan quality and others of saheeh quality.
(7) Imaam Ibn Hajar رحمه الله also presents a definition somewhat between that of Imaam Ibn as-Salaah رحمه الله and that of Imaam Ibn Daqeeq al-'Eid رحمه الله. He says a hasan saheeh hadeeth is one in which there is a difference of opinion concerning the narrators. Some scholars consider them of the first or higher classes while others consider them from the third class, that is, some say they qualify for saheeh hadeeth while others say they only qualify for hasan hadeeth. 
The most important thing to note about ahaadeeth which Imaam at-Tirmidhi رحمه الله labeled hasan saheeh is that they are, at least according to Imaam at-Tirmidhi رحمه الله, acceptable ahaadeeth. Regardless of which of the above definitions is true, this is the point that is most important. Furthermore, many of the ahaadeeth that Imaam at-Tirmidhi رحمه الله called hasan saheeh were also recorded by both Imaam al-Bukhaari رحمه الله and Imaam Muslim رحمه الله which means that they are from among the highest ranking ahaadeeth. 
 See Imaam Jalaalud-Deen as-Suyooti رحمه الله, Tadreeb ar-Raawi fee Sharh Taqreeb an-Nawaawi (Beirut: Daar al-Kutub al-'Ilmiyyah, 1979), vol. 1, p. 161
 Ibid, p. 162.
 Shaykh Ahmad Shaakir رحمه الله, al-Ba'athul Hatheeth Sharh Ikhtisaar 'Uloomul Hadeeth (Cairo: Daar at-Turaath, 1979), pp. 36-37.
 Ibid, p. 36
 Tadreeb ar-Raawi, vol. 1, p. 164.
 Commentary on the Forty Hadeeth of Imaam an-Nawawi رحمه الله, vol. 1, pp. 473-475