One of the most fundamental things to learn when beginning the fish hobby is the nitrogen cycle.

The nitrogen cycle (in the aquarium) is the process which involves bacteria breaking down waste.

Firstly, you have the introduction of waste. Fish poop, uneaten food, decaying plants, dead fish, dead inverts, etc. cause a buildup of ammonia. In the aquarium you need an ammonia level of 0. In a tank with an acidic pH (under 7) it will become ammonium, which is less harmful (about as bad as nitrate). However, this will stall the nitrogen cycle from happening. (more water changes will be needed)

Second, the bacteria known as Nitrosomonas 'eats' this ammonia and creates nitrite, another harmful substance, though not as bad as ammonia. In the aquarium you need a nitrite level of 0.

Nitrobacter bacteria then will consume the nitrite, turning it into nitrate, which is not as harmful as nitrite. In the aquarium, you want a nitrate level under 40, ideally under 20.

To reduce nitrates in the aquarium, plants will help, but you will need to do water changes. Plan a schedule for them, such as week-to-week or twice a week. You can also do them more infrequently.

Anaerobic conditions (deep under sand) will also reduce nitrates but this is very difficult to achieve in the aquarium.

Now how do we cycle an aquarium? Well, you can either do it by way of a fish-in cycle, with a hardy fish such as zebra danios or a fishless cycle, by adding ammonia manually or by letting fish food rot (I wouldn't recommend doing that though).

Several useful products are available which contain the living bacteria needed to cycle an aquarium, the best of which (in my opinion) are Tetra Safe Start (TSS), TSS Plus, or Seachem Stability. To monitor the levels of ammonia, nitrite, nitrate, pH, hardness, or anything else, I would recommend API liquid test kits. Test strips are inaccurate.

The cycle typically takes 2-6 weeks.