Can animals, like monkeys, be homosexual?
Cases of homosexuality have been described in numerous species, including primates, and the wikipedia link posted by @shigeta lists and describes some of them.

It could be difficult to untangle the difference between an animal not being able to identify sexes very well, or, deliberately mating to anyone and those actively making a choice to engage in intercourse with a same sex individual. 

I there are some species where sexes are very difficult to tell apart so they may just go around having sex with anything that looks like the right species because the mating is relatively low cost so doesn't matter if the partner was male or female. This is one such example in sea snails:


  "In a laboratory mate-choice experiment, male N. radiata preferred to
  mate with females, indicating precopulatory sex identification. They
  copulated with males, however, at the frequency of 37%, perhaps
  because of sex misidentification."


Other likely candidates are those with little sexual dimorphism and those with sneaker males that mimic females.

I have also heard theoretical discussions (oh the joys of lunchtime in an evolutionary biology department!) which suggest it may be a way of individuals (particularly males) practising or keeping up fitness relating to sex or stimulating better sperm production etc.

Once you decide what constitutes "a homosexual animal" then it is perhaps easier to examine. I would say a homosexual animal is one that mates exclusively (or highly preferentially) with same sex individuals when given the choice.

Models of genetic and epigenetic inheritance have been used to suggest just how homosexuality could evolve and you can read more about it in this Biology.Stackexchange post.

So it seems that monkeys, and other animals, have been documented to exhibit homosexuality, reasons for them to do so could exist, and it could be a heritable trait which persists despite seemingly strong negative selection.