Okay, the first picture (plant on the right) is a Dendrobium. The second
picture is a Phalaenopsis. They need very different culture from one
another. After going back and re-reading your original post, I'm betting you
have a Dendrobium.

Kenni mentioned keikis. I have Dendrobiums which throw off keikis all the
time, so that could be what you;re seeing. I'd wait to see what develops.
When you water/fertilize, be sure to include those growths.

Let us know what happens!

Diana