What is the conspicuous part of a fern plant called?

The conspicuous part of a fern plant is called the sporophyte. A sporophyte is a mature fern that produces spores.

The sporophyte consists of three major parts: the fronds (leaves), the rhizome and the sporangia. A mature fern creates spores that are released from sporangia. The sporangia in ferns are typically aggregated into the lines of fern fronds.

In the sporophyte phase of fern reproduction, the spores divide mitotically, producing gametophytes. During fertilization, the union of two gametes produces a diploid zygote. The zygote divides mitotically and forms a new sporophyte.

Ferns release spores into the wind. If the spores land somewhere suitable, they eventually grow into a gametophyte.