For deliveries in mammals, I refer to the article about giving birth to that other mammal: mankind. It runs parallel to similar donkeys and rabbits.
Mammals generally can give quite independently birth instinctively, without echoes or maternity.
If they have enough space and materials, most mammals will make a kind of nest before they start labor (by scraping with the hooves on the ground and walk laps in the litter).
nieuwlevenBut because the selection in pets and livestock in recent centuries was done by man and not by nature, there is always selected in many species and cultivated according to many rapidly produced (meat, heavily built, large litters) benefits. The ever larger and heavier animals and offspring can cause a difficult birth.
Most mammals eat the placenta after birth. In domestic swine this is curiously for an omnivore, usually not the case.

In pigs, horses, cows (and dogs) you usually know exactly when you have to breed them, and therefore when to expect the birth. (see <Gestation periods>)
In sheep flocks it is common to buckle on a harness with crayon (colored chalk) to the buck. This is a system of leather straps that hold a block crayons on the chest of the buck. You may have different colors (yellow, green, blue, red) for every week. You can see on the back of the ewes when they are colored, and are thus mated. It makes it easier to count sheep. And the days they have to give birth.

Give pregnant animals plenty of and good food and drinking water.
Make sure, especially by the time they have to deliver - the stable is clean and has a thick layer of clean bedding.
Give them space: usually they try to segregate somewhat for birthing. Provide fencing and gates in the stable (so that other animals not disturb).
Follow the swelling of udder and vulva (also redder and slimy). Look regularly how wide opening they have.
Always have soap, water, clean towels and disinfectant (iodine, for the umbilical cord), petroleum jelly, long gloves ready to use.

If you don’t expect any complications, let no strangers in the stable. Provide a familiar and peaceful surroundings. (No visitors, no dogs...)

Each animal is an investment, and each pregnancy a risk. Both for the mother and the newborn. Go gain experience with colleagues. Or ask an experienced colleague to help. Make sure you have phone numbers of vets on hand if needed.
Do not assume that everything always runs smoothly.
You can, before you expect a delivery, discover in the morning a young stillborn, or a mother is even dead. Keep your animals observing as well as possible, and ask help in time.

In most cases, you will need to do anything at all. The umbilical cord breaks off by itself after birth. The dam will vigorously lick the newborn clean and dry (which gives less loss of heat) and take care.

In an ordinary delivery, the chest will have been significantly compressed in the birth canal. By this the optionally present fluid and mucus in the respiratory tract is largely been removed, while in the expansion of the chest almost automatic air is sucked. In a cesarean this does not happen. Remove mucus (which is sterile, without infecting!) and bravely, strongly rub dry.

First the fruit bladder appears. Usually follows the lamb in less than half an hour. In a breech, as the hind legs come first, you will always have to help by pulling on it. A 2nd and 3rd lamb follow at intervals of 10 to 20 minutes. The afterbirth follows within 1 to 6 hours.
I have always buried it in the manure- or compost heap.
Keep an eye on it that the mother lets drink the newborn. For every lamb that first milk (colostrum) is required, and it drinks allready within the hour! A rejected young dies. Sometimes it still works with some assistance and follow-up. Alternatively you can milk the ewe and giving lamb the bottle, or use a stomach tube.

Provide a lot of fresh litter when an animal will farrow. Check her every 4 hours.

Colostrum is the milk the ewe produces up to 18 hours after birth. It contains important nutritional value with antibodies for the newborn and provides energy to keep the lamb warm.

They should receive adequate colostrum within 30 to 60 minutes after birth. Strip the ewe's teats to remove the wax plugs that often may obstruct the teats. So you’re sure the ewe has milk. (Eventually help the lamb.)

Grafting a disowned lamb and let it drink with another mother (e.g. having a stillborn lamb) can sometimes be done (by the smell of the dead lamb or the (captured) amniotic rubbing extensive on the abandoned animal). For a ewe with 3 or 4 lambs and little milk it may also be indicated on getting used to another dam.
There's a lot involved, and expert help so quickly also makes a difference between life and death.
Fortunately, young animals are much stronger and more independent than human babies.

In cows, the calves are often taken away early. The cows are milked two times a day, and the calves are given a measured amount of milk. A calf that suckles her mother drinks more often and has less digestive problems and diarrhea. And they grow faster.

Please note that this is just a nutshell info. About childbirth for every species you can write books–from breeding to young animal -, especially on gestation, parturition, care and all possible risks and complications. Do not consider yourself an expert after reading a book (let). First go hands-on experience with experts before you go at work yourself.