Tientos is a palo of flamenco, a musical genre originating in Spain in the mid-15th century.  The word derives from the word tentar, meaning to touch, to attempt or to tempt. 

The big 1988 Cinterco Diccionario Enciclopedico Ilustrado de Flamencosays the tiento (it insists on the singular, though it says it’s a plural noun) “is a song with three or four 8-syllable lines, usually followed by one or several 3-line estribillos, whose measure is uniform.  It’s a recent song, dating from the beginning of this century — derived from the earlier tango, which has the same compas, though the tientos is slower, more solemn and complex.

Tientos is a slow rhythm, typically done in a solemn and emotional manner. 

It is majestic, sober and dramatic, with a decidedly ritual air.  

In the cited link to the Flamenco Expierence, the author suggests that tangos and tientos share a similar rhythm. This would appear to be accurate as both the palos have an 8-count compás, the major difference lying in where the accents are placed. Below is a diagram explaining both the compás and palmas of tientos .

Due to the slow nature of tientos, dancers will often do their footwork patterns, llamadas, remates or contestiations in double time and the cantaor will resume the slower tempo with the letra. Observe this in the video below. 

In accordance with the overall feel of tientos, the letras are often solemn. Here are the some of letras of Carmen Linares tientos “Ya No Peino Pa’ti”. 

Si a media noche en tu cama te despierta un viento frio por Dios no te de gindama que son los suspiros mios que a medianoche te llaman

Though the letras of other palos can be solemn as well, it is a trait of tientos letras to be heavier than others. 

Here are more videos to help you get the feel of tientos: 

Camaron y Paco de Lucia 

Cuadra Flamenca de La Lupi 

Carmen Linares “Ya No Me Peino Pa’ti” 

Credit: 

http://www.studioflamenco.com/About_Tientos.html

http://www.flamencoexperience.com/blog/?p=460