The sporophtye is the diploid stage in the life cycle. In comparison, with humans, you and I would be sporophytes.
The Gametophyte is the haploid stage in the life cycle. In comparison, with humans, spermatozoids and ovules are gametophytes.
Note for what follows: "independent" can be understood in terms of "ability to survive". Being independent according to its ability to survive. One might use the word "independent" for other concerns though! For example: "Independent" in the sense "no nutrient exchange between the two"
Therefore, one could say that in humans, the gametophyte is not independent (in it's ability to survive) of the sporophyte. Similarly, in moss (bryophyta), the gametophyte is not independent of the sporophyte because the gametophyte grows on the sporophyte using its nutrients (one could say that the gametophyte parasitize the sporophyte but that would be a bit misleading!). If the sporophyte is independent of the gametophyte (or vice-versa), it means that there is alternation of generations. In other words, sporophyte and gametophyte live separately (like two different individuals).
You can find tons of examples of weird life cycles, in particular in plants, fungi and parasites but not only. And you may find many examples for which the type of sexual reproduction or the type of life-cycle are hard to defined with these relatively simple concepts.
The question of identiy (how separate or genetically different should two individuals be so that we consider them to be two different individuals) is rather a question of philosophy. Such discussions about identity occur when we talk about clones (ramet and genet), about life cycle or about level of organisation/level of selection. The book Major Transition in Evolution (Revisited) provides many of the philosophical discussion on the subject of identity in population genetics.