There are two primary ways that a cell can divide in to two new cells. The first process is called mitosis, and the other is meiosis. In mitosis, a single cell divides in to two identical copies. This is the process for just about every cell in the body, with the exception of sex cells. Sex cell, which include sperm and ova, are formed by a process called meiosis. There are significant differences between mitosis and meiosis.
In mitosis, you start with a cell that has an entire copy of DNA. The DNA is replicated completely, leaving two identical sets. There are distinct stages for mitosis. They are interphase, prophase, metaphase, anaphase, and telophase. Most of the time the cell is in interphase (about 90% of the life cycle). When the cell needs to divide for the organism to grow, chemical signals trigger the start of mitosis and prophase.
All of the stages of mitosis are regulated closely by a complex series of chemical signals. One step must be completed in entirety before the next stage can begin. During mitosis, the nuclear membrane that holds the DNA of the cell will break down. The DNA is then duplicated, lined up, separated in to equal parts, and sent to each side of the cell that is about to split. Of course, the process is far more complicated than this, but that's the essential process.
Meiosis is a completely different process that results in a very different type of cell. Meiosis is a process where the starting cell divides and leaves only half of the starting DNA compared to what it had to start with. In men, the process yields sperm, while in women, you get an egg (or ova). When the egg and sperm get back together, you have a complete set of DNA for the new organism (aka "the baby").
The stages of meiosis are interphase, meiosis I, and meiosis II. Interphase in meiosis is similar to that in mitosis - it is generally a phase in which the cell grows and gathers energy for the upcoming split.
In meiosis I the cell divides and splits a copy of the DNA in to each daughter cell. The cells after meiosis I has 23 chromosomes (this is for humans, other animals have different numbers for this). However, there are two copies of each chromosome.
The second phase in meiosis is meiosis II. At the end of this phase, there are four cells which came from the first parent cell. Each cell has only half of a complete copy of the DNA necessary to live. Meiosis II has a prophase, metaphase, anaphase, and telophase, just like mitosis. The end result yields a different amount of DNA in the resulting cells, but the basic process is very similar.
There are clear differences between mitosis and meiosis. Mitosis is used by non-sex cells to facilitate growth, while meiosis is the process by which gametes (or sex cells) are made. Remember, the process that goes on here is significantly more complicated that this. The details of meiosis and mitosis can fill entire textbooks. As such, this is a mere introduction to some of the major differences between the two processes.