Except in rare circumstances, married Catholics cannot also become priests. It's important to note that this is a discipline, not a doctrine, which means it can be changed in the future. Nonetheless, there are many objections to it - that it's not in the Bible, or that changing it would result in more priests.
Celibacy is indeed mentioned in the Bible. Both Jesus (Mt 19:10-12) and Paul (1 Cor 7:24-35) say that it has its advantages, but is not for everyone. Celibacy allows priests to focus solely on God - the need to take care of a family would hinder the priest's ability to handle his other duties, and the extra salary money needed to raise the family could be a burden on the parish.
- Why Not Married Priests? The Case for Clerical Celibacy - By George Sim Johnson (catholiceducation.org)
- Why Can't Priests Get Married? - Briefly covers all the basics (davidmacd.com)
- Priestly Celibacy - And twelve motives for keeping it (catholicapologetics.info)
- Celibacy for the Kingdom & the Fulfillment of Human Sexuality - A long look at the ultimate meaning of celibacy; by Christopher West (catholiceducation.org)
- Celibacy and the Priesthood - Answers to some Fundamentalist arguments (catholic.com)
- How To Argue For Priestly Celibacy - Answers to six common questions; by Jason Evert (catholic.com)
- 5 Arguments Against Priestly Celibacy and How to Refute Them - Plus five arguments in favor of priestly celibacy (catholiceducation.org)
- Celibacy in the Priesthood - A brief history of celibacy in the Church; by Fr. William Saunders (catholiceducation.org)
- What's the Deal About Legally Married Priests? - How come some Catholic priests are married? By Fr. William Saunders (catholic.com)
- Celibacy Is a Gift - Examines celibate and married life, minimally focusing on the priesthood; by Greg Mockeridge (catholic.com)
- It’s Not About Celibacy - Is there any link between the vow of celibacy and pedophilia? By Deal Hudson (catholiceducation.org)
- Celibacy of the Clergy - From the Catholic Encyclopedia (oce.catholic.com)
- Celibacy is Church Practice, Not Dogma - Verses from the Bible (scripturecatholic.com)
- Doctrines of Demons - Are celibacy and fasting really "doctrines of demons"? By Jimmy Akin (catholic.com)