Your seahorse tank will need some form of Biological Filtration.
Biological filtration uses a process known as the nitrogen cycle - the removal of nitrogenous wastes by bacterial action. Basically this process uses beneficial bacteria to convert toxins like ammonia into relatively harmless substances. All marine tanks need biological filtration. Bio-filters used with some form of mechanical filtration are the best form of filtration for a seahorse tank.
There are quite a few forms of filters that can be used, do some research into the various kinds and choose the one that suits your needs best.
Biological Filtration - The Nitrogen Cycle
Like all fish, seahorses create ammonia as a waste product of digestion. It is also produced through the decay of plant matter and uneaten food from overfeeding. Ammonia is deadly to all fish.
Invisible toxins like ammonia and nitrite are transformed into the relatively harmless substance nitrate by the "Nitrifying" bacteria (Nitrosomonas and Nitrobacter respectively) in a process known as the nitrogen cycle.
These aerobic bacteria (require oxygen to live) attach to surfaces such as gravel, glass, rocks, filter panels, sand and filter media. The more surface area you have the more bacteria can grow.
The Nitrogen cycle can take 3-6 weeks to be working efficiently. It is important to monitor the ammonia, nitrite and nitrate levels during this cycle period.
You will need to test your aquarium every couple of days. Your ammonia levels will be the first to rise, as the first nitrifying bacteria starts acting, you will notice what is called a nitrite spike with high nitrite levels, followed by high nitrate levels.
You will notice these levels lessen and by the fourth or fifth week you should be getting nil ammonia readings (0.0 ppm), nil nitrite levels (0.0 ppm) and readings of > 40 ppm of nitrates.
Once the above readings have been achieved, your bio-filter is active and you can introduce your seahorses to the tank. Without the biological breakdown of toxins the seahorses will literally poison themselves. These beneficial bacteria will grow as your tank ages.
You can start a biological filter going by adding products containing these bacteria, but you should not use fish to start your bio filter. Products like Bio-start or similar are available. Ensure you read all directions carefully.