Chromosomes
Gametophytes are haploid, which means that they only have one copy of each chromosome they inherited from their parent. Conversely, sporophytes are diploid, which means that they have two copies of each chromosome.
Plant Classifications
Plants are classified into four subcategories: bryophytes are primitive plants like mosses and liverworts; tracheophytes are seedless vascular plants like ferns; gymnosperms and angiosperms both have seeds, but angiosperms produce flowers or fruits; gymnosperms have "naked" seeds like pine cones. The sporophyte/gametophyte distinction is relevant to plant classification because all plants have a dominant generation. The dominant generation is the generation that provides nutrition for the non-dominant generation. All plants have a dominant sporophyte generation except for bryophytes. The gametophyte generation is also dominant in most algae.
Cellular Division
Sporophytes reproduce using a process of cell division called "meiosis." Gametophytes, however, reproduce in a process called "mitosis." During mitosis, the cell divides and rearranges its chromosomes resulting in two cells that are genetically identical. Meiosis is more complicated than mitosis and sexual reproduction occurs via meiosis.
Offspring
Using the process of meiosis, sporophytes produce spores. These spores then develop into gametophytes. Conversely, gametophytes produce gametes. In many plants both male and female gametes are produced. These gametes develop into sporophytes, which starts the cycle of alteration of generations over again.