There are a number of ways in which Tenants may seek to have violations corrected. These are applicable both to violations within the apartment itself and to common area violation, those that exist in the public hallways, stairs, basement and other public areas of the building. If there are violations in the public areas it is generally best to include these as well because if everything you are complaining of is behind the locked door of your apartment it is easier for the Landlord to raise defenses related to Notice (you never told me) and Access (I would love to fix that, but you won't let me in). The Landlord is presumed to have notice of the condition of the common areas of the building, as well as things which can be seen from the street (like broken windows if they face the street) and violations of record with the city.

How an Attorney Can Help:

An attorney who has practiced these sorts of cases for many years becomes familiar with what sort of things are violations. I am not a building expert but I have looked over buildings and helped people spot violations they may not have noticed themselves. One example is that if the boiler room is unattended and locked there is required to be a certain sign up. If that sign isn't there, then this is a Class C Violation, which is a major one. There are other examples.
Keep in mind what it is that you expect the Landlord to do and how much it is likely to cost him to do it. This may have a major effect on how hard he will fight.
Many of the options listed below affect each other. Choosing one may preclude you from choosing another.

Complain:

This is the first step, and often required before you take it further. You can do it verbally, but a letter can be exhibited in court.

How an Attorney Can Help:

A letter from an attorney might be taken more seriously by the Landlord. Such a letter will not cost much and if it works you've solved the problem with a minimum of time, money and effort.

Complain to the City:

There are many City agencies which the Tenant can make complaints to, if there is a legitimate problem. The Tenant should keep records of these complaints in order to support a defense of Retaliatory Eviction if that becomes necessary. Housing Preservation and Development (HPD); The Department of Buildings (DoB); Health Department (especially for Rats); Fire Department; Human Rights (Discrimination); Sanitation, Environmental Control Board and others.

The HP Action:

The Housing Preservation, or HP, Action is a proceeding you start in Housing Court against both your Landlord and HPD (Housing Preservation and Development). You are trying to get HPD to force the Landlord to fix the problems. HPD can threaten the Landlord with fines if they don't comply.

How an Attorney Can Help:

An attorney can help you prepare the paperwork for the HP Action and see that everything is listed properly. When the matter gets to court the attorney can try to pressure HPD to pressure the Landlord to make the repairs.

The Rent Abatement:

Rent Abatements are most commonly encountered in the context of a non-payment eviction case. If the building has problems and the Tenant (or Tenant's Attorney) knowns how to proceed it may be possible to obtain an Abatement of Rent either by stipulation or by litigation before the Judge. An Abatement is often expressed as a percentage off the rent, with the percentage varying based upon how bad the place is. Abatements in the 10%-30% range are most common. If your building is a real dive you might get 50%. I have gotten 90%, but it depends on the circumstances. Don't expect a 100% Abatement, these are very rare.

How an Attorney Can Help:

An attorney who is familiar with Housing practice can make sure that the issue is properly raised and followed up on. The attorney can help you understand what sorts of evidence might be required for different conditions and fact patterns. The attorney can attempt to negotiate an Abatement. Landlords sometimes agree to Abatements rather than go before the Judge if they know that the Tenant can show that the building has a lot of problems. If the Landlord won't agree to an Abatement then the attorney could litigate the matter before the Judge and let the Judge decide if an Abatement is proper and how much it should be. The more violations there are, and the more serious they are, the more the Abatement is likely to be, so spotting as many of them as can honestly be found is important.

Decrease in Services Complaints:

These apply only to Rent Controlled and Rent Stabilized tenancies. The Tenant or Tenants complain to DHCR (Department of Housing and Community Renewal) of certain problems and this can affect the rent. There are Building-Wide Decrease in Services Complaint and Individual Apartment Decrease in Services Complaints. These can have different effects on Rent Controlled or Rent Stabilized Tenants. A Rent Stabilized Tenant who prevails on a Decrease in Services Complaint has their rent rolled back to the previous lease and frozen there until the Landlord can get DHCR to issue a Rent Restoration Order.

How an Attorney Can Help:

An attorney could help the Tenant(s) file the Decrease in Services Complaint with the DHCR. An attorney could represent a group of Tenants filing a Building-Wide Decrease in Services Complaint. Rent Stabilized Tenants will not be able to benefit from such a filing unless they sign off on it. The attorney could also help deal with a PAR if one is filed (Petition for Administrative Review, it's like an appeal, many Landlords file these automatically every time they lose).

Repair & Deduct:

Most New Yorkers have heard of Repair & Deduct. It is a process whereby the Tenant can pay for repairs which the Landlord has refused to make and Deduct the cost from the rent. This could be a very practical solution for certain situations. Typically these are things which are simple and inexpensive to correct and located within the apartment. A related remedy is where the Tenants pay for heating oil or electricity service (for the common areas) and deduct that.
It is important that the Tenant not get too ambitious and start ripping out walls or interfering with Building-Wide systems like plumbing and electrical and violating permit and licensing requirements. Such things, and less, could be grounds for eviction. You'll be amazed at how the Landlord's Attorney can wax eloquent about how deeply concerned the Landlord is for the safety of his Tenants, when you're the one who caused a building code violation.

How an Attorney Can Help:

Yes, Repair & Deduct is legal BUT there are REQUIREMENTS about EXACTLY how you go about implementing this. An attorney can advise the Tenant of what those requirements are so that the Tenant might have a chance of actually making the Deduction portion of Repair & Deduct actually work. Most Tenants simply try doing the Repair & Deduct themselves without following the proper procedure, wind up in a non-payment case and then (even if they can get an Abatement on other matters) can't make the Repair & Deduct stick because they didn't do it right. The result is repair, but no deduct.

Article 7A Administrators:

Some buildings are in really bad shape. They are in such bad shape that the cost of fines, rent abatements and decrease in services orders are only a tiny fraction of what it would take to actually do the work. These are the buildings where you are most likely to go so far as to seek the appointment of an Article 7A Administrator. Firstly, this is decidedly a group thing. Certain percentages must be met for this to go anywhere, and that can be tough to get. Lots of people may complain, but may also be afraid to sign off on papers against the Landlord. Linguistic and racial division among the tenants may have to be overcome and there may be divisions between rent regulated and unregulated tenants.
It is also important that the Administrator be someone you can trust.

How an Attorney Can Help:

The appointment of an Article 7A Administrator can sometimes be the solution for a really troubled building. If nothing else has worked and the Landlord totally refuses to do anything it may be that the Tenants need to take action before the building gets so bad the City declares it unsafe. Most Landlords will fight this to the best of their ability. .
This list is not exhaustive. There are Rent Strikes. You can sue the Landlord for breaching the warranty of habitability. If a prior court order has been violated a motion for contempt can be made. There are other possibilities.