Here's the first view. Remember that these are single-celled organisms called dinoflagellates. Each individual is ~1-1.5 mm across.
You can see that they're rounded and shaped somewhat like balloons. The outer surface of the cell is soft.
With a closer look, and when viewed at the right angle, you'll notice that there is a strong ventral groove giving Noctiluca a pinched or kidney-shaped appearance (someone else said it reminded them of a lily pad).
Within the ventral groove there is a tiny flagellum (not visible in these images), a small tooth (also not visible), and a large striated tentacle. The tentacle is easy to see on several individuals in the next two images.
Inside each individual you can see the cytoplasm concentrated in thin, interconnecting strands that extend to the edge of the cell. These strands are visible in all of the photos, but are more distinct under greater magnification (see below).
At the top of the image above, you can also see one individual undergoing binary fission — it's splitting into two.
Noctiluca uses its long tentacle to capture food and sweep it into a slit within the ventral groove. They engulf their food (e.g., diatoms and other dinoflagellates) and store it in temporary compartments called vacuoles. If you scan these images, you'll see food particles stored inside the cells. Most of them look pink in these images (the color depends on what Noctiluca has eaten).
Noctiluca itself is mostly clear, so you may be wondering why the dense patches observed in the ocean are so orange. The color of the patches probably comes from the pigments (e.g., carotenoids) contained within the food that Noctiluca ingests!
The Smithsonian's National Museum of Natural History has summarized information about Noctiluca and provides several helpful photographs and illustrations. Go to their marine dinoflagellates web page and then click on the link at the top where it says "Plate 30, Figs. 1-4."