You should hear back within 2-3 months of your application. You should hear back in early spring — as early as January and no later than April. Some programs will invite you for a faculty interview or campus visit before accepting or declining the offer of admission.
Keep in mind that, unlike undergrad admissions, departments may make PhD admissions incrementally over a period of many weeks. Individual professors review files at their own pace (and often at the last minute).
If you hear another student has been admitted to a program, don’t assume you won’t be admitted. And if you are admitted later, don’t assume that you were seen as less desirable.