Re: How do non flowering plants reproduce?

Nonflowering plants include both gymnosperms (conifers, ginkgo, cycads, etc.) 
and seedless plants. Gymnosperms reproduce by seeds in a similar manner to the 
flowering plants so I'll concentrate on the seedless plants such as ferns and 
mosses. 

All plant life cycles have an alternation of generations between a sporophyte 
and a gametophyte. In seedless vascular plants, such as ferns and horsetails, 
the sporophyte and gametophyte are separate organisms that look very different. 
The fern sporophyte is the familiar large leafy organism. The fern gametophyte 
is a flat, heart-shaped structure less than an inch across. The fern sporophyte 
produces dust-like spores. A spore develops into a gametophyte containing male 
and female parts. A sperm swims to the egg in water on the gametophyte surface 
and combines with it to produce a zygote. The zygote develops into a sporophyte 
and eventually crushs the gametophyte as it grows.

In nonvascular seedless plants, such as mosses and liverworts, the gametophyte 
is the dominant part and bears tiny sporophytes on it.

For further details consult an encyclopedia or biology or botany textbook.

Reference

Young, P. G. 1982. The Botany Coloring Book. New York: Barnes and Noble.