I am frankly a bit surprised to find that this is the English term. I thought is was simply "jumps", but apparently not: the regulations of the FEI (Fédération Equestre Internationale) use the term "jumping efforts" or just "efforts" throughout.

The English version of the FEI regulations is the official version which takes priority over all other versions. I became aware of this through a Argentine Spanish translation of the FEI Eventing rules ("Reglamento de Concurso Completo de Equitación") which states: "La traducción se ha ajustado, en lo posible, a la versión original de la FEI (en inglés) que es prioritaria sobre el Francés."
http://www.fedecuarg.com.ar/completo/Reglamento06/reglamento...

This Spanish version refers on a number of occasions to "esfuerzos de salto"; for example:

"2.1 Si dos o más esfuerzos de salto situados próximos entre sí están diseñados como una prueba integral, se les designa como “elementos” de un solo obstáculo con un solo número. Cada elemento se marcará con una letra diferente (A, B, C, etc.) y se deberá superar en el orden correcto." (Artículo 531 – OBSTÁCULOS)

The FEI Regulations in English use the term "jumping efforts":

"2.1 OBSTACLES COMPOSED OF SEVERAL ELEMENTS - If two or more jumping efforts, sited closely together, are designed as one integral test, they will be designated as "elements" of a single numbered obstacle. Each element will be marked with a different letter (A, B, C, etc.) and must be negotiated in the correct order."
http://www.fei.org/sites/default/files/file/DISCIPLINES/EVEN...

And so it is throughout the Jumping and Eventing rules. For example:

"ARTICLE 16 OBSTACLES
1. At Championships and CSIOPs the course shall consist of 10 to 12 obstacles requiring no more than 15 efforts. The maximum number of combinations shall be one double and one triple or three doubles."

"4.3.18. The number of obstacles in any course may not exceed eight in total, that is, a maximum of ten efforts. Two doubles or one triple may be used."
http://www.fei.org/sites/default/files/file/DISCIPLINES/JUMP...

"4. Number of Jumping Efforts
4.1 THE NUMBER OF JUMPING EFFORTS shall be within the limits shown in Annex 3b.
4.2 TO ARRIVE at the number of jumping efforts, the efforts on the route expected to be taken by the average horse shall be totalled."

"2. General
The obstacles will be standard Show Jumping obstacles. There will be 10 to 13 in number, with 13 to 16 efforts maximum depending on the level of the event as per limits shown in Annex 4."
http://www.fei.org/sites/default/files/file/DISCIPLINES/EVEN...

The Pony Club (UK):
"Obstacles
Number of jumping efforts"
http://www.pcuk.org/v1/index.php?option=com_uhp2&Itemid=86&t...

United States Equestrian Federation:
"JP122 Jumper Courses.
1. The term Obstacles refers to either single obstacles or combination obstacles. All obstacles are numbered numerically. Combination obstacles are those consisting of two or more jumping
efforts, as indicated with the letters “A,” “B,” “C,” etc."
http://www.usef.org/documents/ruleBook/2012/17-JP.pdf

"The course was designed by Rick Cram, and the first round track featured 10 obstacles and 11 jumping efforts."
http://www.millsfamilyfarm.com/ridingLessons-HorseTraining.p...

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Note added at 8 hrs (2012-02-07 18:28:53 GMT)
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A quick word on the alternatives. "Jumps" is not wrong, it seems to me, since "esfuerzos"/"efforts" refers to the number of times the horse is expected to jump in the round. The thing is that (1) it is ambiguous, or could be, since an obstacle is often referred to as a jump (eg. water jump), and a double, which is two "efforts", could be referred to as one jump; and (2) apparently it's not the word they actually use for this in show-jumping circles.

As for "attempts", I must admit that was the first thing that came into my mind to explain the discrepancy between 13 "obstáculos" and 16 "esfuerzos", but if you think about it, it can't mean that. "Attempts" refers to what happens when the horse refuses an obstacle (pulls up short), or runs round it. The rider is (normally) allowed a second attempt. But in theory this could happen at every obstacle: 13 obstacles would be max. 26 attempts (though if it did, the rider would pick up loads of faults and have a disastrous round). There is nothing in any of the rules to suggest that there is a maximum number of attempts per round; there is just a limit of two attempts per obstacle. So "attempts" is definitely wrong here.