We all tend instinctively to shy away from the literal translation, but here I think it's right. "Objetos de apuesta" are simply things on which you bet, or, in this case, are allowed to bet.

It's drawing a distinction between the type of sport and the type of competition or event on which you can bet (like, say, you can bet on horseracing but not dog fighting, or national football competitions but not local ones -- these are hypothetical examples, of course), on the one hand, and then the type of things -- the type of outcomes -- on which you can bet within each "actividad deportiva", on the other. The latter are called "hechos o acontecimientos [...] que podrán ser objeto de cada clase de apuesta". So for example you might be allowed to bet on a given horse to win, but not to bet on a given horse to fall at the third hurdle, or something. That is the kind of distinction being made between "evento" and "objeto".

The corresponding terms in English seem to be "betting events" and "betting objects":

"Betting Events
BettingExpert Owl Sports betting continues to grow more and more popular and betting and sport enthusiasts worldwide follow the major sporting events religiously – and with good reason! Here at BettingExpert we try our best to accommodate all your betting interests and make sure you get a competitive edge when betting on events like Wimbledon, Tour de France, Super Bowl, golf and tennis."
http://www.bettingexpert.com/events/

"The range of declared betting events upon which Tabcorp (NSW Division) and licensed bookmakers are permitted to offer betting is determined by the Minister for Gaming and Racing under section 18 of the Racing Administration Act. "
http://www.olgr.nsw.gov.au/racing_sports_betting.asp

Betting object:
"The first thing that a client has to make is to choose a betting market, a team or a separate player. Then he/she has to analyze a definite set of data to take the right decision on a betting object."
http://www.onlinebettingplanet.com/online-betting-football.h...

"The most common way of betting is to pick an object based on genuine belief of a certain outcome. First pick the object based on belief, then check the prices offered. This how the average punter operate.
Let it be said: Use of statistical analysis should not be the only criteria when choosing betting objects. It should only be used as additional info to a genuine belief. Only use statistics when you are in slight doubt about the object and as a supplement to objects you've singled out through criterias mentioned further down on this page."
http://www.bettingadvice.com/showInfo.php?id=20

The concept of "bet types" or "types of bet" is pretty close to this, but not quite the same thing. A bet type is, for example, a proposition, an each-way bet or an accumulator: it's to do with the mechanics of the wager, if you like. But a betting object is the thing on which you are betting.

--------------------------------------------------
Note added at 5 days (2011-12-23 19:41:42 GMT) Post-grading
--------------------------------------------------

Many thanks, Rich, and have a great Christmas and New Year yourself.