For my wife's deliveries, the common options available were a narcotic (morphine usually, but I understand others use demerol) or epidural.
My wife's family has a history of epidurals flat out stopping labor, which requires pitocin or another labor stimulant, and the labors can take longer, so she's refused to do that each time. The other reason she has avoided epidurals is that she likes to move around during labor, and labor in different positions, which she's been told is difficult to do with an epidural. Perhaps things have changed on that front, though.
Two of the deliveries used morphine, in both cases where the water already broke, and we tried other stimulation, but labor wasn't progressing as quickly as desired. So she took the morphine along with the pitocin.
In one of those cases, the baby was delivered within 4 hours of administering the morphine, which caused some concern as the baby's body isn't developed enough to filter it out of their system, and it can cause breathing problems. They had me rubbing and squeezing his feet to make him cry for some time and they were satisfied that he wouldn't require time in the NICU (Neonatal intensive care unit). He was very, very sleepy, though.
Lesson one - if you use a pain killer other than an epidural, it has to be administered early enough in labor that the mother's body can filter it out and the baby doesn't have issues.
For last child we went to the hospital earlier than the middle three, with enough time to have the morphine again. We brought our wii fit (exercise video game) and passed the time playing with that. Turns out hula hooping really speeds up the labor! It's also very uncomfortable for the mother, but, I suppose, so is labor*. We even got the midwife to hula hoop for a bit. This baby was delivered well after the four hour limit, but still had problems breathing and spent nearly a full day in the NICU. They did xrays and double checked things, but nothing seemed amiss - just needed a bit more time to get used to air. He just turned one and is fine.
For middle three we waited at home until she could stand it no more, then went to the hospital, because laboring in a hospital is harder than at home for us. In all these cases, however, we waited too long for painkillers, and each child was delivered in 1-2 hours after arriving at the hospital.
Lesson two - For us, laboring at home is easier, and results in a natural birth because we already knew we weren't going to do an epidural, and it was too late for other painkillers.
In all deliveries she started out having contractions 5 minutes apart - we didn't get the early warning 15 minute contractions others might experience. In one case she labored at home until transition - that was an interesting ride to the hospital. We usually have to monitor the strength of the contractions to know when to go in - when she couldn't walk or talk through them, or leaned on something it was definitely time to go, and by that time contractions are coming 2-3 minutes apart.
However, the three without any pain medication were wide awake and had great APGAR scores both initially and at the five minute mark. Much better than the two delivered with pain medication. Each delivery is different, though, so we can't draw any significant conclusion from my anecdotal experiences. In any case, all five children are as healthy and intelligent (insofar as we can tell) as the others, so I can't discern any long term effects.
Lesson three - Pain medications, for us, have an initial negative consequence on the baby. There's a stark difference in terms of alertness, activity, and both APGAR scores between those born with pain medication and those without.
We didn't look into other forms of pain medication, and so have no experience or knowledge of others.
We did use a lot of non-medicinal pain relief techniques, but that's another question altogether...
*I did not tell her this at the time. During labor my communications are generally, "Yes dear," "Right away dear," "You're doing great!" and counting. Pretty sure that is my instinctual self preservation kicking in.