I got this plant in May of last year and within two weeks all of the blooms had fallen off. I didn't worry because I was told that would happen periodically but it would bloom again in a couple of months. As you can see it still appears to be healthy, the leaves continue to grow, it started off with just two and now has five beautiful leaves. The roots look good too. I just don't know why it will not bloom. What am I doing wrong?

Next time if you could post this question in the NGAs Orchid Forums you may get more responses.
You were told by who? But don't worry, the plant looks fine. In general, they bloom once a year. They start showing new flower spikes in the fall and bloom November through February. Yours is a teenager. Mature plants can have spikes with branches and upwards of 15-18 flowers. Each flower lasts about a month or so and they bloom consecutively so by the time the first bud opens and dies and then the last one does, you could be looking at 3-4 months of flowers.
Make sure that the pot it is in has drainage and it gets a bright spot without any direct sun!
It likes 55-80 degrees. If it flowers in a room that is too warm, it can shorten the life span of the flowers.
Please look through our forums/ threads on orchids especially the one on Supermarket orchids/Phalaenopsis. Lots of good information in these threads. Welcome on board!

Your plant does look very healthy but hopefully there are many large drainage holes in the bottom of that pot to allow for proper air circulation around the roots and good drainage. Here's the thread BigBill is referring to with information and growing tips regarding Phalaenopsis: The thread "🗣 Starting out with Phals/Supermarket Orchids 🛎 ✍️" in Orchids forum

Here are links to the American Orchid Society Culture Sheets for Phal's:
Novice growers: http://www.aos.org/orchids/cul...
Regular culture sheet: http://www.aos.org/orchids/cul...

Does the plant pot have drain holes? Typically in fall/winter I give mine a slight chill, not COLD but a 10-15 degree F. drop in night temps. This seems to initiate spiking for me.