In October 2012, when the full-time missionary age requirement was reduced for both men and women, President Thomas S. Monson reiterated the call to missionary service, for men, women, and seniors:
We affirm that missionary work is a priesthood duty—and we encourage all young men who are worthy and who are physically able and mentally capable to respond to the call to serve. Many young women also serve, but they are not under the same mandate to serve as are the young men. We assure the young sisters of the Church, however, that they make a valuable contribution as missionaries, and we welcome their service.
We continue to need many more senior couples. As your circumstances allow, as you are eligible for retirement, and as your health permits, I encourage you to make yourselves available for full-time missionary service. Both husband and wife will have a greater joy as they together serve our Father’s children.
So the call to missionary service is:
to young men: a commandment from the Lord. For practical and legal reasons, they must be out of high school, age 18-25 (inclusive), worthy, and physically/mentally/emotionally able. Men who may serve but choose not to don't lose privileges, only blessings. Their missions are 2 years (and 4 days, technically).
to young women: not required or commanded, but their service is certainly welcome. They must be at least 19 years of age (no upper age limit). Their missions are 18 months.
to senior members: senior couples and individuals may serve if they choose, but it's not uncommon for a local leader to call them into a meeting and issue a mission call, knowing the couple or individual is able to serve. Their full-time missions are typically mostly self-funded and offer them a great degree of options because of their seniority and circumstances. Their missions can vary in length according to their situation.
Years ago, to a group of young men, the President of the Church, Spencer W. Kimball, said:
"The question is frequently asked: Should every young man fill a mission? And the answer has been given by the Lord. It is 'Yes.' Every young man should fill a mission. ...
"... Every man should also pay his tithing. Every man should observe the Sabbath. Every man should attend his meetings. Every man should marry in the temple"
("When the World Will Be Converted," Ensign, Oct. 1974, p. 8)
So the commandment for young men to serve a mission is as universal as others in the LDS faith such as attending Church and tithing.
There's a whole article by President Kimball that would be worth reading on this matter.
Elder Rasband of the Presidency of the Seventy, gave a talk a few years ago in a General Conference, wherein, he described the process of full-time mission calls, how they are issued, and why it is a divine commandment.
The whole talk gives a great, little-known insight into issuing calls to missionaries, but he concludes with this, in a more direct answer to your question:
I suppose many of you young men have asked yourselves that same question. Here is the Lord’s answer: "And now, behold, I say unto you, that the thing which will be of the most worth unto you will be to declare repentance unto this people, that you may bring souls unto me, that you may rest with them in the kingdom of my Father." (Doctrine and Covenants 15:6; 16:6.)
At this time in your life, a mission call from the Lord, my young friends, is the most important work that you can do. Prepare now, live righteously, learn from your family and Church leaders, and come join with us in building the kingdom of God on earth—accept your divine appointment in "so great a cause."