An apostrophe as you're using it indicates possession. If you were to use "dog's" with an apostrophe "s", you would be saying that a dog was in possession of something. For example, "the dog's bowl." In this case, the dog owns the bowl or possesses the bowl. Without the apostrophe, it becomes a plural or more than one dog. For example, if you said "who let the dogs out?" you would be asking about more than one dog.
If you wanted to show possession by more than one dog (a group of dogs), you would put the apostrophe after the "s" since you'd still have to show a plural. For example, "the dogs' bowls" would indicate that more than one dog possesses a bowl.
If the word you are trying to use for possession ends with an "s" (for example, Mr. Jones), you would technically have to put an apostrophe after the "s" in Jones, then add another "s" after that. For example, "Mr. Jones's bowl." But really it's fine to just write this without the final "s".