What you have is a Phaleanopsis. While it probably won't have an actual name, the flowers are the harlequin (spotted) variety.
They like medium light and are pretty easy to care for. Medium light would be an east window with a bit of morning sun for instance, or under a grow light. Too much light will cause the leaves to get yellow, too little will cause them to become dark green.
The green houses grow them in plastic sleeves in tightly packed moss, that usually doesn't work so well for home growing. It looks like there is some bark in there and that the roots are currently in good shape. If you are careful with watering - get it well watered then wait until it is really pretty dry before you water it again - you should be able to keep it in the plastic pot until its done blooming. Re-potting stress would probably cause it to lose the flowers. At least you can see the roots through the sides of the pot, that's helpful. As long as they are fat and green (when wet, whitish when dry), they are fine.
When you re-pot an orchid and you use different media (which is pretty much always the case when you get one commercially grown), the roots that are already there are adapted to the media the plant is in, they won't do well and tend to gradually die off. The new roots growing will adapt (to the point of having a different structure even!) to the new media, its just something they have to go through. Generally I put my Phals in coarse bark with some sphagnum moss in it. It is pretty dry at my house, low humidity, and the moss keeps a bit more moisture for them. Phals like good air circulation in the roots, though they need to be snug enough not to wobble. When I go to pot them up, I soak the bark overnight first, so it will absorb moisture. I often water them by soaking the whole plant for a while (from 15 mins to an hour, depending on whether I remember to move them!), then wait a week or even two to water again. My Phals are used to this so they do alright with it. If I have a new one or a small one, I watch it more closely and water more often (as needed).
Hopefully this will give you somewhere to start. As with all internet advice, I recommend waiting for several people to chime in and sifting through what they say while considering your own circumstances. We all do it a bit differently depending on our conditions. Pretty plant and welcome to the forum!