Artificial birth control is morally wrong because it misuses the sexual act by removing the procreative aspect.
Sex is both procreative and unitive. It is meant to bring life into the world, and to unite a husband and wife as one flesh. Many people want to use sex for the pleasure it brings, but remove the life giving aspect. This degrades sex to less than God meant it to be.
Planning your family is necessary. The Church does not teach that we should have tons of children regardless of financial or psychological abilities. However, we should approach sex, not as a right, but as the sacred act it is.
If a couple needs to avoid having a child for good reason, they can avoid sex on the few fertile days of the month. Abstaining from sex does not alter the sex act. In fact, most couples abstain regularly...even those using artificial birth control.
The average married couple has sex twice a week, that's 8 times a month. A couple using Natural Family Planning, who needs to avoid pregnancy, and has sex every day they can, would have sex at twice that average, sixteen (or more) days a month.
Natural Family Planning, is 99% effective when followed properly. It is healthier and promotes discussion about sex and family within the marriage. It also allows for God to be a part of your marriage and your family.
http://news.bbc.co.uk/2/hi/health/6375261.stm
Hormonal birth control, such as the pill or IUD, can cause abortions. Women on these birth control devices, may get pregnant and not know it. The newly conceived baby cannot implant in the uterus and is flushed from it's mother's body. The mother never even knows that her baby was killed.
Ultimately, a couple is responsible to plan their family. Abstaining from sex is not the same as rendering the sex act artificially sterile. The Church has always taught that sex is a procreative act, by divine design and we have no right to change that.
http://www.catholiceducation.org/articles/sexuality/se0002.html
http://www.ccli.org/
http://www.catholic.com/tracts/birth-control
ADDED: - it's obvious from some of the answers that people have been conditioned to believe that not using artificial birth control automatically means having lots kids. This is the mentality of society. No one considers abstinence and self-control as an option.