The best answer to this question can be found on lds.org. Here are a couple of quotes: 
"Human life is a sacred gift from God. Elective abortion for personal or social convenience is contrary to the will and the commandments of God. Church members who submit to, perform, encourage, pay for, or arrange for such abortions may lose their membership in the Church." 
 
"In today's society, abortion has become a common practice, defended by deceptive arguments. Latter-day prophets have denounced abortion, referring to the Lord's declaration, "Thou shalt not . . . kill, nor do anything like unto it" (D&C 59:6). Their counsel on the matter is clear: Members of The Church of Jesus Christ of Latter-day Saints must not submit to, perform, encourage, pay for, or arrange for an abortion. Church members who encourage an abortion in any way may be subject to Church discipline." 
 
"Church leaders have said that some exceptional circumstances may justify an abortion, such as when pregnancy is the result of incest or rape, when the life or health of the mother is judged by competent medical authority to be in serious jeopardy, or when the fetus is known by competent medical authority to have severe defects that will not allow the baby to survive beyond birth. But even these circumstances do not automatically justify an abortion. Those who face such circumstances should consider abortion only after consulting with their local Church leaders and receiving a confirmation through earnest prayer." 
 
"When a child is conceived out of wedlock, the best option is for the mother and father of the child to marry and work toward establishing an eternal family relationship. If a successful marriage is unlikely, they should place the child for adoption, preferably through LDS Family Services." 
 
The answer to the second question - Can someone become a Mormon who has had an abortion? is:  
 
Yes, it is definitely possible for someone who has had an abortion to become a member. It is not automatic that somone can. The interview for baptismal worthiness is more deliberate and permission to be baptized will need to be escalated to a higher level of church authority than is normally required. 
 
As heinous as abortion is - the Lord does allow for deep and significant repentance and forgiveness.

The Church of Jesus Christ of Latter-day Saints does not condone or allow its members to support, fund, promote, endorse, or have an abortion except in very limited situations with the council of your local priesthood leadership (called a Bishop)these situations are if the child is conceived due to rape, incest, or if the health of the mother is at severe risk.  
 
Like previously mentioned these circumstances do not automatically approve an abortion. One should seek confidential council from there Ward Bishop and pray to receive the conformation from the Holy Ghost before making a decision as permanent as abortion.  
 
 
The Church has a extensive adoption program and encourages people consider the option of putting a child up for adoption as the make these decisions.  
 
If a person had an abortion performed in the past this does not bar them from becoming a member of the Church. It is something that is asked during the baptismal interview and can have an impact but in most cases it does not prohibit a person from being baptized.