Sporophyte generation produces spores that grow into a(n) _________?

This can be very confusing, so let me divide it up for you.  There is a term called "alternation of generations", which involves both sexual and asexual reproduction for some plants, fungi, and protists. 

The first cycle is the one you are describing in your question.  A sporophyte is a cell with a diploid number of chromosomes.  It was created by the fertilization of a sperm with an egg cell, resulting in a fertilized egg called a zygote.  The zygote has the full count of chromosomes, making it diploid in number (2n). 

The sporophyte will undergo meiosis, which is a process to reduce the chromosome count by one half, in an effort to produce spores.  The spores, remember, are haploid, meaning they only have half the number of chromosomes required in the cell (n). 

The word you are searching for is gametophyte.  The spores will produce the next cycle, which is the production of a gametophyte from the spores.  The resulting gametophytes are haploid in number (n), so when they undergo mitosis, which is normal cell division, they are only producing half the required chromosome count. 

The gametophytes produce gametes, which are the sperm and egg cells required for sexual reproduction.  When the sperm fertilizes the egg, we are right back to the beginning, where we started with a full count of chromosomes, which is diploid (2n).