There are a number of options.
Most crematoriums either have a pet service that they will provide or can refer you to a crematorium that will provide you with the service. To me this is the way I take handle my pets who are too dear to consider buring in an unmarked grave, or just disposing of otherwise. There are places online that have containers designed to hold the ashes of your pets, and some crematoriums have packages that include a nice container.
Another option is a pet cemetary. These tend to be a more expensive option but, you can get a marker and dedicated place for your beloved pet to rest eternally.
If you have a back yard and it is not prohibited by your local and state laws(it is by some so you will want to check) you can choose to bury your pet in your yard, or that of a friend(with their permission). I personally have a friend who has a pasture that he has buried several of our friends dogs. If you choose this make sure you bury your animal at least 3 feet deep to deter scavengers from digging your pets remains up.
And another option is to dispose of the remains in the trash. The pet's body is biodegradible and unless your disposal company prohibits this, it makes an easy disposal option. I would recommend putting the body into a plastic trash bag to avoid odor from the docomposition and to respect your animal and those people who will collect your trash.