Why is it a requirement for Mormon men to go on missions, but not women?
First of all, if a Mormon decides not to go on a mission, he won't be in trouble. It is highly encouraged, but the Church understands that people have different levels of interest, commitment, and ability. So if you decide not to go, you won't be kicked out, disciplined, or anything like that.
However, a mission is a great opportunity. I went on a mission to Japan, where I shared the gospel with thousands of people. It was an amazing experience, and I would go again in a heartbeat if the occasion came up.
It also helps to understand that Mormon men aren't all asked to serve missions. For example, "Mark" on mormon.org writes:
...I was born with a disability and as such I was unable to serve a full two year mission. I had a desire to serve but it wasn't required of me. I later had opportunities to share The Gospel of Jesus Christ with my friends and with fellow Christians who lived in my home town.
When he writes, "it wasn't required of me," Mark probably means that his bishop or someone else explained that he could be of service in other ways.
Now, what about women? Women are not required to go on missions, but they can do so if they'd like to. Here is a quote from the Church's Young Women Sunday course instruction manual:
Unmarried women age twenty-one and older may also serve full-time missions. However, young sisters should not feel obligated and should not be urged unduly to serve full-time missions. A mission should not interfere with a young woman’s opportunity for marriage.
The Church wants to leave women the opportunity to be married. This isn't surprising, since the Church puts great emphasis on the importance of the family. My own wife did choose to serve a mission, and she's glad that she did. We both felt better prepared for our marriage as a result of the lessons we learned by being full-time servants of Christ, far away from home.