Ok. Gas & air is a mix of nitrous oxide (laughing gas) and oxygen. It can give good pain relief if used properly (deep breaths, starting at the very beginning of the contraction and then stopping as it eases off) but most people don't use it effectively so it doesn't work so well. It can make you feel sick, can make you vomit profusely, can make you feel spaced out/numb, can make you feel sleepy.
An epidural is a small plastic tube which is placed in between the bones of your spine into a space around the layers of tissue surrounding your spinal cord. Local anaesthetic is injected into it and this blocks the nerves carrying the pain sensation. It also blocks nerves which carry temperature and can block the nerves which carry the motor (movement) signals. The big advantage is when they work, they work well - once established, they take all the labour pain away. This can be needed in various cases - for example, where the baby is back to back labour can be more painful and go on longer. Good pain relief allows you to rest and be ready for pushing. It may also stop you pushing too soon if your cervix isn't fully dilated but the baby is low down and making you want to push (pushing too soon makes things worse).
The downside to epidurals are the risks and side effects. The biggest risk (1:10 or so) is it not working. Not all do and some have to be put in a second time if the first doesn't go into quite the right spot. You can get a very bad headache if the epidural needle makes a hole in the dura (causes a leak of spinal fluid) - about 1:100 quoted risk, more common in difficult insertions (very fat, very thin, very mobile). The headache is rarely a serious complication but can require another epidural to patch up the hole. Much rarer risks include nerve damage, infection/abcess formation or blood clots pressing on the spinal cord. Side effects - low blood pressure (makes you feel sick), inability to feel to push (more likely to need assisted delivery eg forceps/ventouse), loss of motor function (so can't walk around with one on the whole although some places offer mobile ones), changes in the baby's heart rate, inability to feel to wee (midwife may need to use intermittent catheterisation to empty your bladder to allow the head to come down).
Epidurals do not make babies sleepier or interfere with bonding or feeding. They also do not influence the likelihood of needing an EMCS (though some evidence is there to show that extreme pain early in labour is a risk factor for EMCS later).
My advice would be to have an extremely flexible birth plan. Don't have your heart set on any one aspect - if you don't get it, it can take a while to come to terms with.