In alternation of generations, a diploid spore-forming cell gives rise to four?
I'm not sure what the reference to alternation of generations is, but the process you are describing with the second part of the question is about the formation of haploid cells with half the chromosome count from diploid cells with the full chromosome count. The cells that are termed haploid are referred to as sex cells and are formed by a process called meiosis.
Virtually all the cells in the human body undergo a process of cell reproduction known as mitosis. Mitosis makes sure each new cell has a diploid number of chromosomes (2n). In humans, this would be 46 chromosomes, matched in 23 pairs. Each cell nucleus, the structure that houses the chromosomes, undergoes a set of phases that copy all the cell structures and make copies of the chromosomes themselves. When all this is complete, the cell spits into two daughter cells, each having the diploid (2n) count of chromosomes, which in humans is 46.
The process of meiosis is for the production of sex cells, sperm for men, eggs for women. The sex cells have to have a haploid (n) count of chromosomes because the egg from the female will be fertilized with a sperm from the male. One haploid + one haploid, during the process of fertilization, will equal one new diploid cell, that will start the process of cell replication on it's own and become a new human being. Meiosis undergoes the same set of steps for the chromosomes as mitosis does, EXCEPT the chromosomes are not copied again. This takes the 46 chromosomes in each of the two daughter cells from mitosis above, and gives two haploid daughter cells for each diploid daughter cell. The result is four haploid cells, each with 23 chromosomes, all ready to combine with another person's 23 chromosomes to make a whole, new person.
Alternation of generations is the term used to describe the life cycle of plants and fungi. In these groups, the life cycle consists of two distinct phases, the sporophyte phase and the gametophyte phase.
The gametophyte phase is a haploid, or 1N, phase, in which each cell contains just one copy of each or the organism's chromosomes. This phase produces the gametes, or sex cells, of the organism. The gametes must find one another and fuse to form a diploid zygote, which is the beginning of the sporophyte stage. During the sporophyte stage, the organism's cells are diploid (2N). Some of these cells will eventually undergo meiosis, which results in one diploid cell giving rise to four haploid spores. The spores are dispersed; they germinate into new gametophytes, and the life cycle repeats.