List two different classifications of drugs used for pain relief during labor and birth, and include potential side effects of the drugs on the fetus.

1. Opioids: These drugs are used to relieve pain during labor and delivery. They work by blocking the perception of pain from reaching the brain, allowing mothers to cope with contractions more effectively. Potential side effects on the fetus include breathing problems, drowsiness and reduced alertness.

2. Nonsteroidal Anti-Inflammatory Drugs (NSAIDs): These drugs reduce inflammation and swelling in the body which can help reduce pain during labor and birth. Potential side effects on the fetus include prolonged pregnancy, premature closure of ductus arteriosus (a fetal blood vessel) and increased risk of bleeding in newborns.