Good aquarium cycle turns Ammonia (NH3) and other organic waste products into Nitrites (NO2) and then Nitrates (NO3) by bacteria.

In any mature aquarium high Nitrates are the primary reason to do frequent water changes. Nitrates are less toxic than Ammonia or Nitrites, but they are still toxic to humans, animals and fish and are a primary cause of water pollution [Note : US EPA's maximum allowable nitrates in tap water is only 10 ppm].

Nitrates are harmles for freshwater fishes if the level below 40 mg/liter (ppm). Nitrates are also the primary cause of unwanted algae growth in an aquarium!.

How to remove Nitrates?
Biological filters only convert organic matter into Nitrates which then accumulate in the aquarium, Mechanical /physical filters only remove larger particulate matter, and Protein Skimmers do not remove any nitrates.

These are comparisons of different methods of nitrate reduction:

Frequent Water Changes
The most common way to control the nitrates in aquariums is to do frequent water changes of 25% to 50% every week or two.

You should pay attention to fluctuating water parameters such as pH and temperature that are associated with water changes which can cause undue stress on your organisms.

Concerns:
• It takes a tremendous amount of time, especially with larger aquariums.

• If you use Tap Water, the water will contain a significant amount of trace elements and potentially harmful chemicals such as chlorine, heavy metals, phosphates, and even more nitrates! If your tap water is hard or alkaline then it shouldn't be used for water changes without any treatment.
Therefore, it is wise to check & filter tap water before using it, you can use Water Purification filters such as Reverse Osmosis / Deionizer filter, UV Sterilizer and Ozonizer .
Or you can use Water Treatment Products such as Water Conditioner or Ammonia & Phosphate Removers as additives.

Note:

Deionization or Reverse Osmosis filtration of tap water nitrates will help if your tap water contains significant levels of nitrates.
Reverse Osmosis strips everything out of the water-the bad, but also the good. Healthy minerals are eliminated as the natural levels in water are altered and the water is now acidic!

• If you use Distilled Water (that has pH near 7.0!), this will likely increase the amount of time you spend doing water changes and over time will cost a great deal. The costs increase even more if you have a saltwater tank.

Note:
Distilled water is water that has many of its impurities removed through distillation. Distillation involves boiling the water and then condensing the steam into a clean container.
Distilled water purchased from a store will have a pH of less than 7.0, often nearer 6.0. This is because after distillation it has been exposed to air, and air contains CO2 which readily dissolves in distilled water and causes a reduced pH.
Only Freshly boiled distilled water (and protected from CO2 during cooling) will be pH 7.0.

• Bottled water / Drinking water. An ideal pH level for drinking water is 9.5 (the higher the pH level, the ORP (antioxidant level), and mineral content in water, the more benefits the body will receive).
But some tests reveal that most of the best-known bottle water brands in the world had a pH level below 7.0, meaning they actually are acidic.
The age of bottled water affects its pH level, as do changes in temperature and exposure to air.

Chemical Nitrate Removal Products (Additives)
Concerns:
Chemical Nitrate Removal Products are just nutrients that you add directly to your tank. It is an uncontrolled reaction that can be very dangerous and have an adverse affect on your organisms.

Nitrate Removing Filter Media
Concerns:
Nitrate removal media will work a bit like a very small denitrator which can not be fed nutrients, they may work a little but they will not be very effective.
They only prolong the time allowed between water changes by keeping nitrates LOW, therefore water changes are still absolutely necessary!

Deep Sand Bed Method
A deep sand bed is designed to cultivate anaerobic bacteria in the bottom layers of sand, converting nitrate to nitrogen gas to remove toxic nitrates.
It's usually not being located in the main tank, but in another container, be it a large bucket, sump, or refugium. It's much easier to take it off-line than to clean out an in-tank deep sand bed if something goes wrong.
This method has been successfully used by aquarists for decades to reduce tank maintenance in Saltwater aquariums, it operates on the same principal as the denitrator.

Concerns:
• It can be complicated to set up and requires a very thick substrate which is itself extremely costly. If this substrate is disturbed it can release a large volume of anoxic water into the tank which can be dangerous.
• It's not nearly as efficient as a denitrator as there is no way to provide nutrients to the bacteria and thus are only suitable for tanks with very few fish.

Refugiums / Vegetable Filter
A refugium is basically a slightly smaller aquarium system than the main aquarium which only contains plants (such as Mangroves) and macro algae. It's an excellent way to control nitrates in a tank and has been successfully used in many Saltwater aquariums.
Refugium can also incorporate a Deep Sand Bed for additional biological filtration capacity.

• In the earlier days, Caulerpa is common algae for refugiums as a nitrate absorber because of its rapid growth under relatively adverse conditions.
Caulerpa was the first, popular algae used in refugiums only because the availability of alternate genera of algae in the hobby was limited.
• "Vegetable Filter" is simply the cultivation of a fast-growing yet stable plant or algae species that competes for (excess) nutrients that would otherwise feed a nuisance algae growth or some other pest organism. Besides being fast-growing and stable, a good algae for refugiums should also be non-aggressive

Concerns:
• Need an additional space, plumbing skills, Costly. It's not for beginners.
• Many introductions of invasive Caulerpa to the wild are thought to have occurred via aquarium dumping. Caulerpa is now illegal to own or sell in Californa and other states may soon follow suit. Caulerpa is very aggressive and sometimes toxic to fishes or invertebrates. For many aquarists, the risks of keeping large portions of Caulerpa far outweigh the benefits.
• "Spaghetti algae" Chaetomorpha and "Ogo" Gracilaria are two of the very best macroalgae for nutrient export in "vegetable filter" style refugiums.

Sulfur Denitrification / Sulfur Nitrate Filter
Sulfur Denitrification is a method that has been used for years in commercial water treatment.

In Sulfur Nitrate Filter the effluent is contains large amounts of H2SO4 (sulfuric acid), and SO4 (sulfate).
You must have a separate reactor with carbonate to neutralize the acid (pH buffering). The resulting effluent is still very high in sulfates and they will accumulate in the tank.

Concerns:
• According to the EPA, high sulfates in drinking water can cause laxative effects.
• It is possible that it may also have adverse effects on sensitive fish.
• The costly sulfur and carbonate media must periodically be replaced.
• Sulfur nitrate filters only harbor a specific species of anaerobic bacteria

Coil Denitrator
A coil denitrator is nothing more than a cylinder with a coil of tubing and some bio-balls. This device works, and achieves the same denitrification results as the more complex and costly commercial units, but much more easily, as well as naturally.

Concerns:
It's has limited nitrate removing capacity and the anaerobic bacteria can't be fed nutrients. It's only suitable for smaller aquariums.

Denitrator (Nitrate Removal Filter)
This type of filter is also known as a denitrifier, biodenitrator, anaerobic biological filter, or simply a nitrate filter.

From my experience, you should try the Aquaripure Nitrate Filter. It's a good product that will surely save your time and money, eliminate nitrates, control algae, reduce maintenance (it will greatly reduce the need for water changes in your fish tanks) and reducing the stress on your aquarium fish, all at once!

The Aquaripure filter is comprehensive biological filter which will completely remove all organic matter and nitrates in an aquarium. It contains at least dozens of different species of both aerobic and anaerobic bacteria and will completely break down all organic matter that passes through it into gasses that escape into the atmosphere.
This makes the Aquaripure filter a comprehensive bacterial filter capable of reducing organic solids and Ammonia (as a common Biological filter) and as well as just Nitrates (Denitrator). I think it can replace other biological filters soon!.

The chemical reaction in the Aquaripure produces Water, Carbon Dioxide and Nitrogen gas.

• Internal Surface Area

To be truly effective and to work in a wide variety of tank set-ups and conditions, the internal surface area of the denitrator has to be greatly increased. The large internal surface area that makes them effective!.

The problem with many of other less expensive denitrators is they are too small or can't be fed nutrients. While they may remove some nitrates, the less expensive and Do-It-Yourself types are poorly designed and too small.

What you need is a robust denitrator which has an enormous internal surface area, and works in all tank conditions and set-ups and will eliminate nitrates in nearly any aquarium.
Aquaripure filters all have a tremendous internal surface area for denitrifying bacteria to grow and thrive, much more than any "coil" denitrator or the other competition. For example, it would take more than ten "coil" type nitrate filters to equal the nitrate removing power of an X-Large Aquaripure. Moreover, many other nitrate filters are vastly overrated in terms of their capacity.

• Price, Safety, Easy Maintenance & Operating Cost

The other effective denitrators that are available cost much more than the Aquaripure, are complicated to maintain, and often require the purchase of expensive proprietary additives or substrate.

In contrast, Aquaripure's denitrators are reasonably priced, safe, effective, and easy to set up and maintain. Unlike other filters in which you have to replace the internal media, you will never have to change anything on the Aquaripure. There are no filters to change and no mess to deal with.
There is a "feeding" tube on all of Aquaripure's filters (except the Mini) that is very easy to use. You simply inject sugar water or alcohol into the Aquaripure once a week. Feeding and adjusting the Aquaripure only costs pennies and will only take seconds of your time. Aquaripure's filters are designed to be easy to setup and maintain, and will give you many years of service.