It appears you have a specimen of Phalaenopsis sp. also known as moth orchid.
Your particular specimen, judging by the photo included in your question, has strong new root growth. This is encouraging - you have been doing something right.
From Wikipedia...
Phalaenopsis are among the most popular orchids sold as potted plants,
owing to the ease of propagation and flowering under artificial
conditions.
In nature, Phalaenopsis thrive in temperatures around 20 to 35 °C (68 to 95 °F), but are adaptable to conditions more comfortable for human habitation in temperate zones 15 to 30 °C (59 to 86 °F). Note that at temperatures below 18 °C (64.4 °F) overwatering causes root rot. Phalaenopsis requires high humidity (60–70%) and low light of 12,000 to 20,000 lux.
Phalaenopsis drop their blooms as spring turns to summer and then begin their growth season. Expect them to spike again with the onset of the chill of early fall/autumn and bloom again in the following late winter or spring.
From http://www.ranwild.org/Phalaenopsis/module/introduction/intro1/Eenvironhead.html
Phalaenopsis is an epiphytic plant with roots that run on tree trunks
and branches. It is different from parasitic plants in that genus
Phalaenopsis does not drain the mother plant of nutrients. Epiphytic
orchids have a symbiotic relationship with one type of fungi and form
orchid mycorrhiza in the root cells. The orchid provides the fungi
with sugars produced by photosynthesis, while the fungi give the
orchid the nitrogen compounds (mainly nitrogen and phosphorus)
produced when the sugars are broken down by the fungal enzymes. The
orchid produces the organic substance (carbohydrates and vitamins) by
itself through photosynthesis, which is the difference between it and
parasitic plants.
Phalaenopsis species grow even with the bare-root hanging in the
air if enough moisture can be provided. In nature, Phalaenopsis
exposes their roots to air and adheres to the host trunk tree to
obtain moisture from temporal rainwater that runs down the host tree.
The high humidity at night in the rainforest minimizes the moisture
loss from their leaves and assists the respiration.
How does this information translate into plant care? Read on...
We need to attempt to replicate the plants natural habitat as best as practically possible.
I strongly recommend that you do not use potting mix to grow this plant. This will at the very least stagnate the growth of your orchid and most likely lead to its early demise. Generally a very free draining “orchid mix” potting medium will be suitable. The contents will vary greatly depending on where you are located in the world - the medium may contain course bark, perlite, vermiculite, different types of moss, course coconut fibre, charcoal, cork. Many people grow Phalaenopsis in only sphagnum moss, although a common mistake is to pack it too tightly around the roots, which creates an environment with too much moisture and not enough air.
A small pot is all that is required. As the plant grows and the root system develops, you may need to repot your orchid. Repotting is more to ensure the medium is of good quality and less to accommodate the plant, which in nature literally grows on the surface of the bark of the host tree.
Phalaenopsis growing on cork with a little sphagnum moss against the roots
Cold conditions slow growth and can damage flowers, so keep the plant away from:
- glass, which can get cold overnight;
- air-conditioners and draughts, which cause accelerated evaporation and low humidity.
To achieve the required humidity, stand the pot on a tray of pebbles and fill the tray with water. The pebbles keep the pot out of the water and allow the potting medium to completely drain after each watering.
To achieve the required light, place the plant in a brightly lit warm spot out of direct sunlight.
To achieve the required nutrient supply, feed with a soluble orchid fertiliser per the recommendations on the packet. Separately and in between these “feeds”, apply a tonic of half strength liquid soil bacteria and liquid seaweed / kelp extract.
To start, water more in the growing season (every 7 days) and less in the dormant season (every 14 days). Your careful observation and discretion is required however so keep an eye on the potting medium and adjust the frequency so it does not dry out, although recognise that it is important for the plant’s health to experience wet and dry conditions. Remember that prolonged damp conditions around the roots might lead to root rot.