The NYCHA representative will give you a date for the repairs. If it is an emergency (flood, carbon monoxide, etc.) they should come within 24 hours. Most simple repairs are done within 48 hours. However, NYCHA might give you a repair date that is several months away. If this happens you should consider filing a court case.
For repairs scheduled within a week, NYCHA will have someone from maintenance come to your apartment to look at the conditions. They may or may not make the repairs. The maintenance worker may ask you to sign a paper saying the repairs were done or that someone from maintenance visited your apartment. Make sure you read everything before you sign it.
If NYCHA does not come to your apartment at the scheduled time, you can contact the management office to ask what happened.
If NYCHA does not make the repairs at all you can file a court case to get repairs. This type of case is called an HP action.