This is a pretty easy concept. There are two ways you can approach this.
One way is to sneak up to the animal in the wild when it is hungry and toss some food at it and back away, you will need to do this multiple times and eventually it will be tamed.
The second and more carefree way to do this is to lure the animal into a pen when it is chasing you and throw a bundle of food in the pen. Every time the animal is hungry it will eat and eventually be tamed. On the pictures to the right you can see the stack of mushrooms in the pen and the boar I want to tame acclimatizing. As long as you stay away from the pen, the boar will continue to eat and tame. This is the preferred method because you can go about your day-to-day tasks. When your animal has tamed, a notice will pop up on the screen alerting you that your animal has been tamed.
Once tamed you can pet your animal, hearts arise, and you will see that they love you.
Foods for Taming and Breeding:
Boars: Carrots, Red Mushrooms, Turnips, Berries.
Wolves: Raw Meats.
Lox: Barley and Cloudberries.
Once you have tamed two adult animals you can start breeding them. I did this by keeping the animals in a small pen and kept food in the pen for them at all times. When both of the boars I wanted to breed ate, they would go close to each other and light pink hearts would rise above them. This would signal that they are mating. It took a few different tries for them to mate in order to produce piggies.
When a baby is born you will see red around your animals, a baby will emerge, and a squeal will be heard.
Once you have around 5-6 animals you should extend your pen to be larger so your animals can keep breeding. Many breeders are using a smaller pen attached to a bigger pen method: They let the animals breed in the small pen so they are in close proximity to each other and then push the babies out to the larger pen.
Note: The animals must be tame and happy in order to mate. If they are hungry or frightened they will not produce young.
Animals will breed if they are in the biome natural to them. Wolves, for instance, need to be in the mountain biome.